Normalized Solutions for Schrödinger Equations with Stein–Weiss Potential of Critical Exponential Growth

Abstract

In this paper, we focus on the existence of normalized solutions to the following Schrödinger equation with the Stein–Weiss potential

$$\begin{aligned} -\Delta u+\lambda u=\left( I_{\mu }\times \frac{F(u)}{|x|^{\alpha }}\right) \frac{f(u)}{|x|^{\alpha }}, x\in {{\mathbb {R}}} ^2, \end{aligned}$$

where $2\alpha +\mu \le 2$, $0<\mu <2$, $I_{\mu }$ denotes the Riesz potential and $f:{{\mathbb {R}}} \rightarrow {{\mathbb {R}}}$ has critical exponential growth which behaves like $e^{\alpha u^2}$. The solutions correspond to critical points of the underlying energy functional subject to the $L^2$-norm constraint, namely, $\int _{{{\mathbb {R}}} ^2} |u|^{2}\textrm{d}x = a^2$ for $a>0$ given. Under some weak assumptions, we prove the existence of the normalized solution for the equation by developing refined variational methods. In particular, we shall establish two new approaches to estimate precisely the minimax level of the underlying energy functional. As far as we know, our result is the first one in seeking normalized solutions of nonlinear equations involving the nonlocal Stein–Weiss reaction.

1 Introduction

In this paper, we are concerned with the existence of normalized solutions to the following nonlinear Schrödinger equation with the Stein–Weiss reaction under the exponential critical growth case

where $a>0$, $0<\mu <2$, $2\alpha +\mu \le 2$. $I_{\mu }$ denotes the Riesz potential defined by

$$\begin{aligned} I_{\mu }(x)=\frac{\Gamma (\frac{\mu }{2})}{\Gamma (\frac{2-\mu }{2})2^{2-\mu }\pi |x|^{\mu }}:=\frac{A_{\mu }}{|x|^{\mu }}, \ x\in {{\mathbb {R}}} ^{2}\setminus \{0\}, \end{aligned}$$

where $\Gamma$ represents the gamma function, $*$ indicates the convolution operator, F(s) is the primitive of f(s) with that f(s) has exponential critical growth in ${{\mathbb {R}}} ^{2}$. Next, we shall introduce three typical features of this problem to set the tone for the rest of the paper.

1.1 Introduction of Three Typical Features

1.1.1 $L^2$-Constraint

We aim to search for solutions to $({\mathcal {P}}_{a})$ having prescribed mass, the normalized stationary states, whose existence can be formulated as follows: given $a>0$, we aim to find $(\lambda ,u)\in {{\mathbb {R}}} \times H^{1}({{\mathbb {R}}} ^{2})$ solving $({\mathcal {P}}_{a})$ together with the normalization condition

$$\begin{aligned} |u|_{2}^{2}=\int _{{{\mathbb {R}}} ^{2}}|u|^{2}\textrm{d}x=a, \end{aligned}$$

and in this case $\lambda \in {{\mathbb {R}}}$ cannot be prescribed but appear as Lagrange multipliers in the variational approach. This type of problem has important physical significance in Bose–Einstein condensates and the nonlinear optics framework, and the $L^{2}$-norm of such solutions is a preserved quantity of the evolution and the corresponding variational feature contributes to analyzing the orbital stability or instability. Naturally, such problems have attracted much attention in the fields of nonlinear PDEs in the last decades. We give brief introduction of the relative progress with the most general Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda u=|u|^{p-2}u, &{} \hbox {in}\ {{\mathbb {R}}} ^N, \ N\ge 2,\\ \int _{{{\mathbb {R}}} ^N}u^2\textrm{d}x=a, \end{array} \right. \end{aligned}$$

(1.1)

which has been investigated extensively via the variational methods. One can search for the existence of the normalized solutions of (1.1) by considering the critical points of the corresponding energy functional ${\mathcal {J}}: H^1({{\mathbb {R}}} ^N) \rightarrow {{\mathbb {R}}}$ defined by

$$\begin{aligned} {\mathcal {J}}(u) = \frac{1}{2}\int _{{{\mathbb {R}}} ^N}|\nabla u|^2\textrm{d}x -\frac{1}{p}\int _{{{\mathbb {R}}} ^N}|u|^p \textrm{d}x, \end{aligned}$$

under the constraint

$$\begin{aligned} {\mathcal {S}}_{a,N}=\left\{ u\in H^1({{\mathbb {R}}} ^N):\int _{{{\mathbb {R}}} ^{N}}|u|^2=a\right\} . \end{aligned}$$

Generally, the types of such problems are divided into three parts: $L^{2}$-subcritical case, $L^{2}$-critical case, $L^{2}$-supercritical case, after a simple stretching consideration presented by

$$\begin{aligned} u_{t,N}(x)=t^{\frac{N}{2}}u(x), \end{aligned}$$

during which one can find there exists a new $L^{2}$-critical exponent $q^{*}=2+4/N$. Generalizing to more general nonlinear term f, one can conclude that if f admits a $L^{2}$-subcritical growth, i.e., f has a growth $u^{p-1}$ with $p<q^{*}$ at infinity, then ${\mathcal {J}}|_{{\mathcal {S}}_{a,N}}$ is bound below and in this occasion, minimization method is the conventional approach to find normalized solutions, we refer to [12, 31] and the references therein in this aspect. If f admits a $L^{2}$-supercritical growth, i.e., f has a growth $u^{p-1}$ with $p>q^{*}$ at infinity, then ${\mathcal {J}}$ is unbound below on ${\mathcal {S}}_{a,N}$, which implies the traditional minimization method does not work and more efforts are always required in the study of the $L^2$-supercritical case.

One of the groundbreaking pieces of work in the $L^2$-supercritical case is accomplished by Jeanjean [21]. Jeanjean [21] obtained the normalized solution at the mountain pass level of its energy functional under the following conditions:

• (H0) f is odd;

• (H1) $f\in {\mathcal {C}}({{\mathbb {R}}} , {{\mathbb {R}}} )$ and there exist $\alpha ,\beta \in {{\mathbb {R}}}$ satisfying $(2N+4)/N<\alpha \le \beta <2^*=2N/(N-2)$ such that

  $$\begin{aligned} 0<\alpha F(t)\le f(t)t\le \beta F(t), \ \ \ \ \forall \ t\in {{\mathbb {R}}} \setminus \{0\}, \end{aligned}$$

and the existence of ground state solutions was proved if f also satisfies

• (H2) the function ${\tilde{F}}(t):=f(t)t-2F(t)$ is of class ${\mathcal {C}}^1$ and

  $$\begin{aligned} {\tilde{F}}'(t)t>\frac{2N+4}{N}F(t), \ \ \ \ \forall \ t\in {{\mathbb {R}}} \setminus \{0\}. \end{aligned}$$

Condition $\mathrm{(H1)}$ is the general Ambrosetti–Rabinowitz condition, which is benefit for us to obtain bounded Palais–Smale sequences for ${\mathcal {J}}$ constrained on ${\mathcal {S}}_{a,N}$, and Jeanjean developed a novel argument related to the mountain pass geometry for the scaled functional ${\tilde{{\mathcal {J}}}}(u,s):=J(u,s)$ with $s*u(\cdot ):=e^{\frac{Ns}{2}}(e^{s}\cdot )$, which is widely used to find normalized solutions in the $L^{2}$-supercritical case. Based on the above conditions and by applying the fountain theorem to the scaled functional ${\mathcal {J}}$, Bartsch and de Valerioda [5] obtained infinitely normalized solutions. Another variational approach is presented by Bartsch and Soave [6, 7], and it is based on Ghoussoub minimax principle [19]. Applying this abstract minimax theorem, Barstch and Soave established the existence and multiplicity of normalized solutions of equation (1.1) with general nonlinearities f. We also refer to Bieganowski and Mederski [8], which considered the normalized solutions under the $L^{2}$-supercritical case but Sobolev subcritical, they provided nearly optimal conditions in some ways. For more information, please see [8] and its references.

We conclude this part with a brief progress about the critical Schrödinger equation in the sense of Sobolev embedding when $N\ge 3$. In 2020, Soave [28] first considered the Schrödinger equation with Sobolev critical growth:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda u=\mu |u|^{q-2}u+|u|^{2^*-2}u, &{} \hbox {in}\ {{\mathbb {R}}} ^N,\ N\ge 3, \\ \int _{{{\mathbb {R}}} ^N}u^2\textrm{d}x=a. \\ \end{array} \right. \end{aligned}$$

(1.2)

This type of problem is more delicate since they need to analyze how the lower-order term $|u|^{q-2}u$ affects the structure of the corresponding energy functional and to solve the lack of compactness caused by the Sobolev critical growth. According to the findings in [28], Eq. (1.2) has ground state solutions in the $L^2$-subcritical perturbation case $2<q<2+N/4$ and $L^2$-supercritical perturbation case $2+N/4<q<2^*$, respectively, for $\mu a(1-\gamma _q)q<\alpha$, where $\alpha =\alpha (N,q)$ is a specific constant depending on N, q and $\gamma _q=N(q-2)$. We also refer to Wei and Wu [33], Jeanjean and Le [22], Jeanjean et al. [24] which settled several open questions proposed by Soave [28].

1.1.2 Critical Exponential Case

Besides the $L^2$-constraint, another novel feature of equation (${\mathcal {P}}_{a}$) is that functions f(u) have critical exponential growth that is the maximal growth that allows us to treat (${\mathcal {P}}_{a}$) variationally in $H^1({{\mathbb {R}}} ^2)$, which was shown by Trudinger [32] and Moser [26], and it is motivated by the following Trudinger–Moser inequality [11].

Lemma 1.1

(i) If $\alpha >0$ and $u\in H^1({{\mathbb {R}}} ^2)$, then

$$\begin{aligned} \int _{{{\mathbb {R}}} ^2}\left( e^{\alpha u^2}-1\right) \textrm{d}x<\infty ; \end{aligned}$$

(ii) if $u\in H^1({{\mathbb {R}}} ^2), \Vert \nabla u\Vert _2^2\le 1, \Vert u\Vert _2 \le M < \infty$, and $\alpha < 4\pi$, then there exists a constant ${\mathcal {C}}(M,\alpha )$, which depends only on M and $\alpha$, such that

$$\begin{aligned} \int _{{{\mathbb {R}}} ^2}\left( e^{\alpha u^2}-1\right) \textrm{d}x \le {\mathcal {C}}(M,\alpha ). \end{aligned}$$

Inspired by the Trudinger–Moser-type inequality, we can say that a function $f\in {\mathcal {C}}({{\mathbb {R}}} ,{{\mathbb {R}}} )$ possesses critical exponential growth if there exists a constant $\alpha _{0}>0$ such that

Here, we refer Adimurthi–Yadava [36] and de Figueiredo–Miyagaki–Ruf [17] to the readers for more information. Based on the Trudinger–Moser inequalities, many authors considered the existence and multiplicity of weak solutions for the nonlinear Schrödinger equations. We refer to [2, 13, 14] and the references therein for more details about the relative progress in this direction.

1.1.3 Stein–Weiss Convolution Reaction

We briefly recall the related background and some pioneering contributions in this field, and we start with the weighted $L^{p}$ estimates for the fractional integral

$$\begin{aligned} (T_{\mu }\varphi )(x)=\int _{{{\mathbb {R}}} ^{N}}\frac{\varphi (y)}{|x-y|^{\mu }}\textrm{d}y, \ \ 0<\mu <N, \end{aligned}$$

which is a fundamental problem in the field of harmonic analysis and such weighted $L^{p}$ estimates are generated from quite natural phenomena and have practical significance in the large wide of mathematical fields, which can be summarized as that the appearance of some suitable symmetry hypotheses, notably radial symmetry, contributes to improving the classical estimates and some embedding properties of function spaces.

Many mathematicians have studied on the weighted $L^{p}$ estimates for the fractional integral $T_{\mu }$. Historically, Hardy and Littlewood [20] first considered the weighted $L^{p}$ estimates for the one-dimensional fractional integral operator $T_{\mu }$, then Sobolev [29] extended it to the N-dimensional case. Later, Stein and Weiss [30] obtained the following two-weight extension of the Hardy–Littlewood–Sobolev inequality, which is known as the Stein–Weiss inequality.

Proposition 1.2

(Doubly weighted Hardy–Littlewood–Sobolev inequality) Let $t,s>1$ and $0<\mu <N$ with $\vartheta +\beta >0$, $1/t+(\mu +\vartheta +\beta )/N+1/s=2$, $\vartheta <\frac{N}{t'}$, $\beta <\frac{N}{s'}$, $g_{1}\in L^{t}({{\mathbb {R}}} ^{N})$ and $g_{2}\in L^{s}({{\mathbb {R}}} ^{N})$, where $t'$ and $s'$ denote the Hölder conjugate of t and s, respectively. Then there exists a constant $C(N,\mu ,\vartheta ,\beta ,t,s)$, independent of $g_{1}$, $g_{2}$ such that

$$\begin{aligned} \int _{{{\mathbb {R}}} ^{N}}\int _{{{\mathbb {R}}} ^{N}}\frac{g_{1}(x)g_{2}(y)}{|x-y|^{\mu }|y|^{\vartheta }|x|^{\beta }}\textrm{d}x\textrm{d}y \le C(N,\mu ,\vartheta ,\beta ,t,s)\Vert g_{1}\Vert _{t}\Vert g_{2}\Vert _{s}. \end{aligned}$$

For $\vartheta =\beta =0$, it is reduced to the Hartree type (also called the Choquard type) nonlinearity, which is driven by the classical Hardy–Littlewood–Sobolev inequality.

Integrability for integral operators can be quantified using the Stein–Weiss inequality, which is fundamentally determined by the dilation nature of integral operators. Due to its significance in applications to issues in harmonic analysis and partial differential equations, the study of and comprehension of the Stein–Weiss inequality has sparked a growing amount of attention among scholars. We now look at the applications that the Stein–Weiss term has in relation to them. The polyharmonic Kirchhoff equations involving the critical Choquard-type exponential nonlinearity with singular weights were explored by Giacomoni et al. [4]. It is important to note the beautiful work of Du et al. [18], where they investigated the equation as below,

$$\begin{aligned} -\Delta u=\frac{1}{|x|^{\alpha }}\left( \int _{{{\mathbb {R}}} ^{N}}\frac{|u(y)|^{2_{\alpha ,\mu }^{*}}}{|x-y|^{\mu }|y|^{\alpha }}\textrm{d}y \right) |u(x)|^{2_{\alpha ,\mu }^{*}-2}u, \quad x\in {{\mathbb {R}}} ^{N}, \end{aligned}$$

where $2_{\alpha ,\mu }^{*}=(2N-2\alpha -\mu )/(N-2)$. In order to analyze the existence of solutions, study the regularity, and symmetry of positive solutions by moving plane arguments under the critical situation, as well as the results under the subcritical situation, the authors created a nonlocal version of the concentration—compactness principle. Yang et al. [38] achieved the symmetry, regularity, and asymptotic features of the weighted nonlocal system with critical exponents associated with the Stein–Weiss inequality by employing the moving plane arguments in integral form. For other results, we refer to [9, 37, 39, 40] and the references therein.

1.2 Introduction of Our Goal and Main Results

Among the investigations into normalized solutions of the nonlinear Schrödinger equation with critical growth, an new emerging interest is seeking the normalized solutions under the critical exponential growth in the sense of the Trudinger–Moser inequality, which was recently constructed by Alves et al. [3]. Under following hypothesis,

(F2$'$):

$\lim _{|t|\rightarrow 0}|f(t)|/|t|^{l}=0$ for some constant $l>3$;

(F3$'$):

there exists a constant $\mu _0>4$ such that $f(t)t\ge \mu _0 F(t)>0$ for all $t\in {{\mathbb {R}}} \setminus \{0\}$;

(F4$'$):

there exist constants $p>4$ and $\gamma >0$ such that $F(t)\ge \gamma |t|^p$ for all $t\in {{\mathbb {R}}}$;

(F5$'$):

the function ${\tilde{F}}(t):=f(t)t-2F(t)$ is of class ${\mathcal {C}}^1$ and satisfies

$$\begin{aligned} {\tilde{F}}'(t)t\ge 4F(t), \ \ \ \ \forall \ t\in {{\mathbb {R}}} , \end{aligned}$$

they established the existence of normalized solutions to equation:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda u=f(u), &{} \hbox {in}\ {{\mathbb {R}}} ^2, \\ \int _{{{\mathbb {R}}} ^2}u^2\textrm{d}x=a. \\ \end{array} \right. \end{aligned}$$

(1.3)

Their result reads as follows in this topic.

Theorem 1.3

([3, Theorem 1.2]) Assume that f possesses critical exponential growth and satisfies (F2$'$)-(F4$'$). If $a\in (0,1)$, then there exists $\gamma ^*(a)>0$ such that (1.3) has a radial solution for all $\gamma \ge \gamma ^*(a)$, where $\gamma$ is given by (F4$'$), moreover, this solution can be chosen as a positive ground state solution if f also satisfies (H0) and (F5$'$).

Based on the idea introduced by Jeanjean [21], for every $a\in (0,1)$, they constructed a special (PS) sequence $\{u_n\}\subset {\mathcal {S}}_a^r:= {\mathcal {S}}_a\cap H_r^1({{\mathbb {R}}} ^2)$ such that

$$\begin{aligned} \varphi (u_n)\rightarrow c_{\gamma }^{\infty }(a)>0, \ \ \varphi |_{{\mathcal {S}}_a^r}'(u_n) \rightarrow 0\ \ \text{ and }\ \ \vartheta (u_n)\rightarrow 0, \end{aligned}$$

where the mountain pass level $c_{\gamma }^{\infty }(a)$ depends on $\gamma$ and $\varphi : H^1({{\mathbb {R}}} ^2)\rightarrow {{\mathbb {R}}}$ is given by

$$\begin{aligned} \varphi (u) = \frac{1}{2}\int _{{{\mathbb {R}}} ^2}|\nabla u|^2\textrm{d}x-\int _{{{\mathbb {R}}} ^2}F(u)\textrm{d}x, \end{aligned}$$

on the constraint ${\mathcal {S}}_a$ which is defined by

$$\begin{aligned} {\mathcal {S}}_{a}=\left\{ u\in H^1({{\mathbb {R}}} ^2):\int _{{{\mathbb {R}}} ^{2}}|u|^2=a\right\} . \end{aligned}$$

As a consequence of the Pohozaev identity (see [21, Lemma 2.7]), any solution u of $\varphi$ exists in the Pohozaev manifold given by

$$\begin{aligned} {\mathcal {M}}_{a} =\left\{ u\in {\mathcal {S}}_a: \vartheta (u)=0\right\} , \end{aligned}$$

where $\vartheta$ is called the Pohozaev functional defined by

$$\begin{aligned} \vartheta (u) = \Vert \nabla u\Vert _2^2 -\int _{{{\mathbb {R}}} ^2}[f(u)u-2 F(u)]\textrm{d}x, \ \ \ \ \forall \ u\in H^1({{\mathbb {R}}} ^2). \end{aligned}$$

Next, let us discuss a few key components of their proofs in the work [3]. By establishing the crucial estimation

$$\begin{aligned} \limsup _{n\rightarrow \infty }\Vert \nabla u_n\Vert _2^2 \rightarrow 0 \ \ \hbox {as}\ \ \gamma \rightarrow \infty , \end{aligned}$$

they could overcome the influence caused by the exponential critical growth. Indeed, as long as $\limsup _{n\rightarrow \infty }\Vert \nabla u_n\Vert _2^2<1-a$, then the following property

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{{{\mathbb {R}}} ^2}|u_n|^s\left( e^{\alpha u_n^2}-1\right) \textrm{d}x =\int _{{{\mathbb {R}}} ^2}|{\bar{u}}|^s\left( e^{\alpha {\bar{u}}^2}-1\right) \textrm{d}x, \end{aligned}$$

is a natural conclusion, since we have the Trudinger–Moser inequality and the compact embedding $H_r^1({{\mathbb {R}}} ^2)\hookrightarrow L^s({{\mathbb {R}}} ^2)$ for any $s>2$. Then they were able to derive the Brezis–Lieb property

$$\begin{aligned} \int _{{{\mathbb {R}}} ^2}\left[ f(u_n)u_n-f({\bar{u}}){\bar{u}}-f(u_n-{\bar{u}})(u_n-{\bar{u}})\right] \textrm{d}x=o(1), \end{aligned}$$

and also the convergence

$$\begin{aligned} \int _{{{\mathbb {R}}} ^2}\left[ f(u_n)u_n-f({\bar{u}}){\bar{u}}\right] \textrm{d}x=o(1), \end{aligned}$$

if $u_n\rightharpoonup {\bar{u}}$ in $H^1({{\mathbb {R}}} ^2)$. In this case, the dealing process of their problem is similar to the one of nonlinearities like $f(u)\sim |u|^{q-2}u$ with $q>2$. After this work, a series of subsequent studies have been done on the existence of different types of normalized solutions to the nonlinear equations in the planar case. Here, we refer to the work by Jeanjean and Lu [23], and Chen and Tang [15] and the references therein for more details.

Motivated by the above works, especially [3], in this paper, we discuss some refined analysis for the existence of normalized solutions to nonlinear Schrödinger equation with Stein–Weiss convolution term. As far as we know, our result is the first one in the fields of seeking normalized solutions involving the nonlocal Stein–Weiss reaction.

To state our results, besides (F1), we make the assumptions on f:

• (F2) $\lim _{|t|\rightarrow 0}f(t)/t^{\frac{4-\mu -2\alpha }{2} }=0$;

• (F3) $f(t)t\ge (6-\mu -2\alpha )F(t)/2$ for all $t\in {{\mathbb {R}}} \setminus \{0\}$;

• (F4) $\liminf _{|t|\rightarrow \infty }\frac{ f(t)}{e^{\alpha _0 t^2}}>0$;

• (F5) there exist constants $M_0>0$ and $\beta _0>0$ such that

  $$\begin{aligned} F(t)\le M_0 |f(t)|, \ \ \ \ \forall \ |t|\ge \beta _0. \end{aligned}$$

Theorem 1.4

Assume that $a>0$, $\alpha <\mu$ and f satisfies (F1)–(F5). Then equation $({\mathcal {P}}_{a})$ has a radial normalized solution. Moreover, for any solution the associated Lagrange multiplier $\lambda$ is positive.

Remark 1.5

We would like to point out that the purpose of both Theorem 1.4 and following Theorem 1.6 is to demonstrate that radial solutions exist for $({\mathcal {P}}_{a})$. As a result, we focus on the space $H_r^1({{\mathbb {R}}} ^2)$ since it compactly embeds in $L^s({{\mathbb {R}}} ^2)$ for all $s>2$ and aids in the recovery of compactness. The solutions in $H_r^1({{\mathbb {R}}} ^2)$ are in reality solutions in whole $H^1({{\mathbb {R}}} ^2)$ according to Palais’ principle of symmetric criticality [35].

Set

$$\begin{aligned} \gamma ^*(a) = \frac{A^{\frac{2p-(6-2\alpha -\mu )}{4}}}{ \sqrt{{\mathcal {C}}(N,\mu ,\alpha )}a^{\frac{4-2\alpha -\mu }{4}}C_{ \frac{4p}{4-2\alpha -\mu }}^{\frac{4-2\alpha -\mu }{4}}}\left[ \frac{2\alpha _{0}}{(4-2\alpha -\mu )\pi }\right] ^{\frac{2p-(6-2\alpha -\mu )}{4}}. \end{aligned}$$

(1.4)

where A is defined in (3.10).

Theorem 1.6

Assume that $a>0$, $\alpha <\mu$ and f satisfies (F1)–(F3), (F4$'$) with $\gamma >\gamma ^*(a)$ and (F5). Then conclusions of Theorem 1.4 hold.

Let us now outline the main strategy to prove Theorems 1.4 and 1.6. Our arguments are based on variational approaches and refined analysis techniques in order to complete the proofs of main results. It is easily seen that solutions of problem $({\mathcal {P}}_{a})$ can be found by looking for critical points of the energy functional $\Phi :H^{1}({{\mathbb {R}}} ^{2})\rightarrow {{\mathbb {R}}}$ given by

$$\begin{aligned} \Phi (u)=\frac{1}{2}\int _{{{\mathbb {R}}} ^{2}}|\nabla u|^{2}-\frac{1}{2}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(u)}{|x|^{\alpha }}\right) \frac{F(u)}{|x|^{\alpha }}\textrm{d}x. \end{aligned}$$

By proposition 1.2, we know the latter term of the right-hand side of equation $\Phi (u)$ is well defined if $F(u)\in L^{t}({{\mathbb {R}}} ^{2} )$ for $t>1$ given by $2/t+(\mu +2\alpha )/2=2$. This means that we must require $F(u)\in L^{\frac{4}{4-\mu -2\alpha }}({{\mathbb {R}}} ^{2})$, which can be guaranteed by (F1), (F2) and the continuous Sobolev embedding $H^1({{\mathbb {R}}} ^2)\hookrightarrow L^p({{\mathbb {R}}} ^2)$, where $p\ge 2$. Furthermore, it is standard to show that $\Phi \in {\mathcal {C}}^1(H^1({{\mathbb {R}}} ^2), {{\mathbb {R}}} )$. More precisely, since the exponential critical growth in ${{\mathbb {R}}} ^{2}$ implies that the energy functional $\Phi$ is no more bounded from below on ${\mathcal {S}}_{a}$, and we shall look for a critical point satisfying a minimax characterization. Here, we introduce the following definition.

Definition 1.7

For given $a>0$, we say that $\Phi$ possesses a mountain pass geometry on ${\mathcal {S}}_a$ if there exists $\rho _a>0$ such that

$$\begin{aligned} {c}(a):=\inf _{g\in \Gamma _a}\max _{\tau \in [0,1]}\Phi (g(\tau ))>\max _{g\in \Gamma _a}\max \{\Phi (g(0)),\Phi (g(1))\}, \end{aligned}$$

(1.5)

where $\Gamma _a:=\{g\in {\mathcal {C}}([0,1],{\mathcal {S}}_a):\Vert \nabla g(0)\Vert _2^2\le \rho _a, \Phi (g(1))<0\}$.

We intend to make use of the above definition to verify the mountain pass geometry of our problem. Besides, we highlight how the Stein–Weiss convolution term, in conjunction with the critical exponential growth, presents some new difficulties to our approach, as well as the accomplishments in our paper, which can be seen below.

• (i) One of the main challenges comes after the appearance of the Stein–Weiss convolution term. We must confirm that the weak limit of the (PS) sequence of energy functional $\Phi$ is, in fact, the solution to the equation $({\mathcal {P}}_{a})$. However, in the case of exponential growth, the appearance of the Trudinger–Moser inequality requires that the critical exponent $\alpha _{0}$ be less than $2\pi$. We notice that Alves and Shen [1] provided a version of proof of such property; here, we would like to show another version of proof based on the condition that $\alpha <\mu$, which is motivated by the method explored in Qin and Tang [27, Lemma 4.8].

• (ii) The work by Alves et al.[3] is significant in the field of normalized solutions in the two-dimensional critical case, and the technique previously described demonstrates that it is possible to control the energy level of the corresponding energy functional arbitrary small by simply taking a large enough value for the parameter $\gamma$. This estimation is one of the most crucial components that cannot be overlooked because we are working with the exponential critical case. The first technique we developed in this instance is to provide an exact lower bound $\gamma$, which is defined by (1.4). We shall provide a precise range of values for $\gamma$ to fulfill the energy estimation in this method.

• (iii) We would like to point out that, in the ordinary methods, one usually could take advantage of the Moser-type functions to pull down the critical value to a particular threshold value. However, it seems like there is no such estimation in seeking normalized solutions of the nonlinear equations except the very recent work by Zhang et al. [41], where they used the traditional Moser-type function, then to perform a stretch to satisfy the constraint mass and then gave the estimation. Here, motivated by Chen et al. [16], we will directly improve the traditional Moser-type function. By the suitable extension of the traditional Moser-type function, we can obtain the test functions in $H^1({{\mathbb {R}}} ^2)$ on the $L^2$-constraint ${\mathcal {S}}_a$. Another highlight in this progress is our assumption with the growth on f at infinity is relatively weak compared to the existing works about the Stein–Weiss convolution term.

• (iv) Our results are superior than those of [3] in that we additionally change the constraint condition utilized in [3] from $a\in (0,1)$ to $a>0$.

Remark 1.8

We emphasize that the ground state normalized solution can be obtained using our work with some additional assumptions. However, in some ways, finding a ground state normalized solution is similar to the process explored as in our paper.

The organization of the remainder of this paper is as follows. In Sect. 2, we shall introduce some preliminary results and establish the mountain pass geometry of the associated energy functional. In Sect. 3, we shall apply two different approaches to give a precise estimation for the mountain pass energy level. In Sect. 4, we shall restore the compactness and prove the existence of normalized solutions of the equation ${\mathcal {P}}_{a}$.

Finally, we introduce some notations that will clarify what follows.

$\bullet$ $C,C_{i},c_{i}\ (i=1,2,...)$ denote positive constants which may vary from line to line.

$\bullet$ For any exponent $p>1$, $p'$ denotes the conjugate of p and is given as $p'=p/(p-1)$.

$\bullet$ $B_{r}(x)$ denotes the ball of radius r centered at $x\in {{\mathbb {R}}} ^{2}$.

$\bullet$ The arrows $\rightharpoonup$ and $\rightarrow$ denote the weak convergence and strong convergence, respectively.

$\bullet$ $L^{s}({{\mathbb {R}}} ^{2})(1\le s<+\infty )$ denotes the Lebesgue space with the norm $\Vert u\Vert _{s}=(\int _{{{\mathbb {R}}} ^{2}}|u|^{s}\textrm{d}x)^{1/s}$.

$\bullet$ $\beta : H:=H^1({{\mathbb {R}}} ^2)\times {{\mathbb {R}}} \rightarrow H^1({{\mathbb {R}}} ^2)$ is a continuous map defined by

$$\begin{aligned} \beta (v, t)(x)=e^{t}v(e^tx)\ \ \text{ for }\ v\in H^1({{\mathbb {R}}} ^2),\ t\in {{\mathbb {R}}} \ \text{ and }\ x\in {{\mathbb {R}}} ^2, \end{aligned}$$

(1.6)

where H is a Banach space equipped with the scalar product

$$\begin{aligned} ((v_1,s_1),(v_2,s_2))_H=(v_1,v_2)+s_1s_2, \ \ \ \ \forall \ (v_i,s_i)\in H, \ i=1,2, \end{aligned}$$

and corresponding norm $\Vert (v,t)\Vert _{H}:=\left( \Vert v\Vert ^2+|t|^2\right) ^{1/2}$ for all $(v,s)\in H$.

2 Preliminary Results

In this section, we give some preliminary results which will be useful throughout the rest of the paper.

Lemma 2.1

(Gagliardo–Nirenberg inequality [34]) Let $q>2$. Then there exists a sharp constant ${\mathcal {S}}_{q}>0$ such that

$$\begin{aligned} \Vert u\Vert _{q}\le {\mathcal {S}}_{q}^{1/q}\Vert \nabla u\Vert _{2}^{\frac{q-2}{q}}\Vert u\Vert _{2}^{\frac{2}{q}}, \end{aligned}$$

where ${\mathcal {S}}_{q}=\frac{q}{2\Vert U_{q}\Vert _{2}^{q-2}}$, and $U_q$ is the ground state solution of the following equation:

$$\begin{aligned} -\Delta U+\frac{2}{q-2}U=\frac{2}{q-2}|U|^{q-2}U. \end{aligned}$$

To deal with the nonlocal type problem (${\mathcal {P}}_{a}$), we also need the following inequality.

Lemma 2.2

(Cauchy–Schwarz type inequality [25, Sect. 5]) For f, $h\in L_{\textrm{loc}}^{1}({{\mathbb {R}}} ^{2})$, there holds

$$\begin{aligned} \int _{{{\mathbb {R}}} ^{2}}\left( \frac{1}{|x|^{2-\mu }}\times |f|\right) |h|\textrm{d}x\le \left[ \int _{{{\mathbb {R}}} ^{2}} \left( \frac{1}{|x|^{2-\mu }}\times |f|\right) |f|\textrm{d}x\int _{{{\mathbb {R}}} ^{2}}\left( \frac{1}{|x|^{2-\mu }}\times |h| \right) |h|\textrm{d}x\right] ^{\frac{1}{2}}. \end{aligned}$$

Finding the bounded (PS) sequence relies on the mountain pass geometry based on Definition 1.7. Especially, verifying the boundedness of such a sequence is not trivial, which needs the information of $L^2$-Pohozaev inequality. Here, we introduce the following:

$$\begin{aligned} \Phi (tu_{t})=\frac{t^{2}}{2}\Vert \nabla u\Vert _{2}^{2}-\frac{t^{\mu +2\alpha -4}}{2}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(tu)}{|x|^{\alpha }}\right) \frac{F(tu)}{|x|^{\alpha }}\textrm{d}x, \end{aligned}$$

(2.1)

and

$$\begin{aligned} \begin{aligned} J(u)=\frac{\textrm{d}}{\textrm{d}t}\Phi (tu_t)\Big |_{t=1} =&\ \Vert \nabla u\Vert _2^2+\frac{4-\mu -2\alpha }{2}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(u)}{|x|^{\alpha }}\right) \frac{F(u)}{|x|^{\alpha }}\textrm{d}x \\&-\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(u)}{|x|^{\alpha }}\right) \frac{f(u)u}{|x|^{\alpha }}\textrm{d}x. \end{aligned} \end{aligned}$$

(2.2)

Following that, we show that $\Phi$ has a mountain pass geometry on the constraint ${\mathcal {S}}_a$, which is as follows.

Lemma 2.3

Assume that (F1)–(F3) and (F5) hold. Then

• (i) there exists $K(a)>0$ small enough such that $\Phi (u)> 0$ and $J(u)>0$ if $u\in A_{2K}$ and

  $$\begin{aligned} 0<\sup _{u\in A_{K}}\Phi (u)<\inf \left\{ \Phi (u): u\in {\mathcal {S}}_a, \Vert \nabla u\Vert _2^2= 2K(a) \right\} , \end{aligned}$$

  (2.3)

  where $A_{K}=\left\{ u\in {\mathcal {S}}_a: \Vert \nabla u\Vert _2^2\le K(a)\right\}$ and $A_{2K}=\left\{ u\in {\mathcal {S}}_a: \Vert \nabla u\Vert _2^2\le 2K(a)\right\}$.

• (ii) $\Gamma _a=\{g\in {\mathcal {C}}([0,1],{\mathcal {S}}_a):\Vert \nabla g(0)\Vert _2^2\le K(a), \Phi (g(1))<0\}\ne \emptyset$ and

  $$\begin{aligned} {c}(a) =\inf _{g\in \Gamma _a}\max _{t\in [0,1]}\Phi (g(t))&\ge \inf \left\{ \Phi (u): u\in {\mathcal {S}}_a, \Vert \nabla u\Vert _2^2= 2K(a) \right\} \\&>\max _{g\in \Gamma _a}\max \{\Phi (g(0)),\Phi (g(1))\}. \end{aligned}$$

Proof

(i) Fixing $\alpha >\alpha _0$, by (F1) and (F2), we know that for any $\varepsilon >0$ and any $q\ge 1$, there exists $C_{\alpha ,\varepsilon ,q}>0$ such that

$$\begin{aligned} |f(t)| \le \varepsilon |t|^{\frac{4-\mu -2\alpha }{2}} + C_{\alpha ,\varepsilon ,q}|t|^q \left( e^{\alpha t^2}-1\right) , \ \ \ \ \forall \ t\in {{\mathbb {R}}} , \end{aligned}$$

(2.4)

moreover, using (2.4), we deduce that for any $\varepsilon >0$, there exists $C_{\alpha ,\varepsilon }>0$ such that

$$\begin{aligned} |F(t)| \le \varepsilon |t|^{\frac{6-\mu -2\alpha }{2}} + C_{\alpha ,\varepsilon } |t|^{\frac{10-\mu -2\alpha }{4}} \left( e^{\alpha t^2}-1\right) , \ \ \ \ \forall \ t\in {{\mathbb {R}}} . \end{aligned}$$

(2.5)

Thus, by Proposition 1.2, Lemma 2.1, and (2.5) and by selecting $\varepsilon$ small enough, we have

$$\begin{aligned} \Phi (v)&\ge \frac{1}{2}\Vert \nabla v\Vert _{2}^{2}-\frac{C(2,\mu ,\alpha )}{2}\Vert F(v)\Vert _{\frac{4}{4-\mu -2\alpha }}^{2} \\&\ge \frac{1}{2}\Vert \nabla v\Vert _{2}^{2}-\frac{C(2,\mu ,\alpha )}{2}\left( \int _{{{\mathbb {R}}} ^{2}}C_{1}\varepsilon ^{\frac{4}{4-\mu -2\alpha }} |v|^{\frac{2(6-\mu -2\alpha )}{4-\mu -2\alpha }}\textrm{d}x\right. \\&\left. \quad + \int _{{{\mathbb {R}}} ^{2}}C_{1}C_{\alpha ,\varepsilon }^{\frac{4}{4-\mu -2\alpha }}\left( e^{\alpha v^2}-1\right) ^{\frac{4 }{4-\mu -2\alpha }}|v|^{\frac{10-\mu -2\alpha }{4-\mu -2\alpha }}\textrm{d}x \right) ^{\frac{4-\mu -2\alpha }{2}} \\&\ge \frac{1}{2}\Vert \nabla v\Vert _{2}^{2}-\frac{C(2,\mu ,\alpha )}{2} C_{2}C_{1}^{\frac{4-\mu -2\alpha }{2}}\varepsilon ^{2}\Vert v\Vert _{\frac{2(6-\mu -2\alpha )}{4-\mu -2\alpha }}^{6-\mu -2\alpha } \\&\quad -\frac{C(2,\mu ,\alpha )}{2}C_{3}C_{1}^{\frac{4-\mu -2\alpha }{2}}C_{\alpha ,\varepsilon }^{2} \\&\left( \int _{{{\mathbb {R}}} ^{2}}\left[ e^{\frac{8\alpha v^{2}}{4-\mu -2\alpha }}-1\right] \textrm{d}x \right) ^{\frac{4-\mu -2\alpha }{4}}\left( \int _{{{\mathbb {R}}} ^{2}}|v|^{\frac{2(10-\mu -2\alpha )}{4-\mu -2\alpha }}\right) ^{\frac{4-\mu -2\alpha }{4}} \\&\ge \frac{1}{2}\Vert \nabla v\Vert _{2}^{2}-\frac{C(2,\mu ,\alpha )}{2} C_{2}C_{1}^{\frac{4-\mu -2\alpha }{2}}\varepsilon ^{2}{\mathcal {S}}_{{\frac{2(6-\mu -2\alpha )}{4-\mu -2\alpha }}}^{-\frac{4-\mu -2\alpha }{2}}a^{4-\mu -2\alpha }\Vert \nabla v\Vert _{2}^{2} \\&\quad -\frac{C(2,\mu ,\alpha )}{2}C_{3}C_{1}^{\frac{4-\mu -2\alpha }{2}}C_{\alpha ,\varepsilon }^{2} \left( \int _{{{\mathbb {R}}} ^{2}}\left[ e^{\frac{8\alpha v^{2}}{4-\mu -2\alpha }}-1\right] \textrm{d}x \right) ^{\frac{4-\mu -2\alpha }{4}} \\&\quad {\mathcal {S}}_{\frac{2(10-\mu -2\alpha )}{4-\mu -2\alpha }}^{-\frac{4-\mu -2\alpha }{4}}a^{\frac{4-\mu -2\alpha }{2}}\Vert \nabla v\Vert _{2}^{3} \\&= \frac{1}{2}\Vert \nabla v\Vert _2^2-C_{4}a^{ 4-\mu -2\alpha }\varepsilon ^{2}\Vert \nabla v\Vert _2^2- C_{5}a^{\frac{4-\mu -2\alpha }{2}}\Vert \nabla v\Vert _{2}^{3} \\&\ge \frac{1}{4}\Vert \nabla v\Vert _2^2- C_{5}a^{\frac{4-\mu -2\alpha }{2}}\Vert \nabla v\Vert _{2}^{3} . \end{aligned}$$

Now, let $0<K$ be arbitrary but fixed and suppose $u,u_0,v,v_0\in {\mathcal {S}}_a$ satisfy that $\Vert \nabla u\Vert _2^2\le K$, $\Vert \nabla v\Vert _2^2\le 2K$, and $\Vert \nabla v_0\Vert _2^2=2K$. From above, when $K>0$ is small enough, we can say that $\Phi (v)>0$. Similarly, we can obtain that $J(v)>0$ and again by selecting $\varepsilon$, K small enough, we have

$$\begin{aligned} \Phi (v_0)-\Phi (u)&= \frac{1}{2}\Vert \nabla v_0\Vert _2^2-\frac{1}{2}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(v_{0})}{|x|^{\alpha }}\right) \frac{F(v_{0})}{|x|^{\alpha }}\textrm{d}x-\frac{1}{2}\Vert \nabla u\Vert _2^2\\&\quad +\frac{1}{2}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(u)}{|x|^{\alpha }}\right) \frac{F(u)}{|x|^{\alpha }}\textrm{d}x\\&\ge \frac{1}{2}K-C_{6}a^{ 4-\mu -2\alpha }\varepsilon ^{2}K-C_{7}a^{\frac{4-\mu -2\alpha }{2}}(2K)^{3/2} \\&= \frac{1}{4}K-C_{8}a^{\frac{4-\mu -2\alpha }{2}}(2K)^{3/2}\ge \frac{1}{8}K. \end{aligned}$$

From above, there exists $K=K(a)>0$ sufficiently small such that $\Phi (u)>0$ and $J(u)>0$ if $u\in A_{2k}$, and (2.3) holds.

(ii) We first prove that $\Gamma _a\ne \emptyset$. Using (F3) and (F5), it is easy to see that

$$\begin{aligned} F(t)\ge F(\beta _0 sign(t))e^{(|t|-\beta _0)/M_0}, \ \ \ \ \forall \ |t|\ge \beta _0. \end{aligned}$$

(2.6)

For any given $w\in {\mathcal {S}}_a$, we have $\Vert tw_t\Vert _2=\Vert w\Vert _2$, and so $tw_t\in {\mathcal {S}}_a$ for every $t>0$. Then (2.1) and (2.6) yield

$$\begin{aligned} \Phi (tw_t)\rightarrow -\infty \ \ \hbox {as}\ \ t\rightarrow +\infty . \end{aligned}$$

(2.7)

Thus we can deduce that there exist $t_1>0$ small enough and $t_2>0$ large enough such that

$$\begin{aligned} \Vert \nabla (t_1w_{t_1})\Vert _2^2=t_1^2\Vert \nabla w\Vert _2^2\le K(a), \ \ \Vert \nabla (t_2w_{t_2})\Vert _2^2=t_2^2\Vert \nabla w\Vert _2^2>2K(a)\ \hbox {and}\ \Phi (t_2w_{t_2})<0. \end{aligned}$$

Let $g_0(t):=(t_1+(t_2-t_1)t)w_{t_1+(t_2-t_1)t}$. Then $g_0\in \Gamma _a$, and so $\Gamma _a\ne \emptyset$. Now using the intermediate value theorem, for any $g\in \Gamma _a$, there exists $t_0\in (0,1)$, depending on g, such that $\Vert \nabla g(t_0)\Vert _2^2=2K(a)$ and

$$\begin{aligned} \max _{t\in [0,1]}\Phi (g(t))\ge \Phi (g(t_0))\ge \inf \left\{ \Phi (u): u\in {\mathcal {S}}_a, \Vert \nabla u\Vert _2^2= 2K(a) \right\} , \end{aligned}$$

which, together with the arbitrariness of $g\in \Gamma _a$, implies

$$\begin{aligned} {c}(a)=\inf _{g\in \Gamma _a}\max _{t\in [0,1]}\Phi (g(t))\ge \inf \left\{ \Phi (u): u\in {\mathcal {S}}_a, \Vert \nabla u\Vert _2^2= 2K(a) \right\} . \end{aligned}$$

(2.8)

Hence, (ii) follows directly from (2.3) and (2.8), and the proof is completed. $\square$

We recall that any solution of $({\mathcal {P}}_{a})$ lives in the $L^2$-Pohozaev manifold given by

$$\begin{aligned} {\mathcal {M}}_a=\left\{ u\in {\mathcal {S}}_a: J(u):=\frac{\textrm{d}}{\textrm{d}t}\Phi (tu_t)\Big |_{t=1}=0\right\} . \end{aligned}$$

Remark 2.4

From $J(v)>0$ when $v\in {\mathcal {S}}_{c}$ and $\Vert \nabla v\Vert _{2}^{2}\le 2K$, we can deduce that for any $a>0$, there exists a constant $\rho (a)>0$, just depending on $a>0$, such that $\Vert \nabla u\Vert _2\ge \rho (a)$ for all $u\in {\mathcal {M}}_a$.

Next, inspired by [21], we consider the following auxiliary functional:

$$\begin{aligned} {\tilde{\Phi }}(v,t)=\Phi (\beta (v,t)) =\frac{e^{2t}}{2}\Vert \nabla v\Vert _2^2 -\frac{e^{(\mu +2\alpha -4)t}}{2}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(e^{t}v)}{|x|^{\alpha }}\right) \frac{F(e^{t}v)}{|x|^{\alpha }}\textrm{d}x.\nonumber \\ \end{aligned}$$

(2.9)

We shall show that ${\tilde{\Phi }}$ also possesses a kind of mountain pass geometrical structure on ${\mathcal {S}}_a\times {{\mathbb {R}}}$. Since the proof is standard, we omit it here.

Lemma 2.5

Assume that (F1)–(F3) and (F5) hold. Let $v\in {\mathcal {S}}_a$ be arbitrary but fixed. Then we have

• (i) $\Vert \nabla \beta (v, t)\Vert _2\rightarrow 0$ and $\Phi (\beta (v,t))\rightarrow 0$ as $t\rightarrow -\infty ;$

• (ii) $\Vert \nabla \beta (v, t)\Vert _2\rightarrow +\infty$ and $\Phi (\beta (v,t))\rightarrow -\infty$ as $t\rightarrow +\infty ;$

• (iii) there exist $s_1<0$ and $s_2>0$, depending on a and v, such that the functions ${\tilde{v}}_1=\beta (v,s_1)$ and ${\tilde{v}}_2=\beta (v,s_2)$ satisfy

  $$\begin{aligned} \Vert \nabla {\tilde{v}}_1\Vert _2^2\le K(a),\ \ \Vert \nabla {\tilde{v}}_2\Vert _2^2>2K(a) \ \ \hbox {and}\ \ \Phi ({\tilde{v}}_2)<0. \end{aligned}$$

Lemma 2.6

Assume that (F1)–(F3) and (F5) hold. Then

$$\begin{aligned} {c}(a)={\tilde{c}}(a):=\inf _{{\tilde{g}}\in {\tilde{\Gamma }}_a}\max _{\tau \in [0,1]}{\tilde{\Phi }}({\tilde{g}}(\tau )) >\max _{{\tilde{g}}\in {\tilde{\Gamma }}_a}\max \left\{ {\tilde{\Phi }}({\tilde{g}}(0)),{\tilde{\Phi }}({\tilde{g}}(1))\right\} , \end{aligned}$$

where

$$\begin{aligned} {\tilde{\Gamma }}_a:=\{{\tilde{g}}\in {\mathcal {C}}([0,1],{\mathcal {S}}_a\times {{\mathbb {R}}} ): {\tilde{g}}(0)=({\tilde{g}}_1(0),0), \Vert \nabla {\tilde{g}}_1(0)\Vert _2^2\le K(a), {\tilde{\Phi }}({\tilde{g}}(1))<0\}. \end{aligned}$$

By the argument explored as in [35], we know that for any $a>0$, ${\mathcal {S}}_a$ is a submanifold of $H^1({{\mathbb {R}}} ^2)$ with codimension 1 and the tangent space at ${\mathcal {S}}_a$ is defined as

$$\begin{aligned} T_u=\left\{ v\in H^1({{\mathbb {R}}} ^2):\int _{{{\mathbb {R}}} ^2}uv\textrm{d}x=0\right\} . \end{aligned}$$

The norm of the ${\mathcal {C}}^1$ restriction functional $\Phi |_{{\mathcal {S}}_a}'(u)$ is defined by

$$\begin{aligned} \Vert \Phi |_{{\mathcal {S}}_a}'(u)\Vert =\sup _{v\in T_u,\Vert v\Vert =1}\left\langle \Phi '(u),v\right\rangle . \end{aligned}$$

As in Jeanjean [21], for every $(u,t)\in {\mathcal {S}}_a\times {{\mathbb {R}}}$, we define the following linear space

$$\begin{aligned} {\tilde{T}}_{u,t}=\left\{ (v,s)\in H:\int _{{{\mathbb {R}}} ^2}uv\textrm{d}x=0\right\} . \end{aligned}$$

We see that ${\tilde{\Phi }} (v,t)$ is of class ${\mathcal {C}}^1$ and for any $(w,s)\in H$,

$$\begin{aligned} \left\langle {\tilde{\Phi }}'(v,t),(w,s)\right\rangle&= e^{2t}\int _{{{\mathbb {R}}} ^2}\nabla v\cdot \nabla w\textrm{d}x+e^{2t}s\Vert \nabla v\Vert _2^2 \\&\ \ \ -e^{(\mu +2\alpha -4)t}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(e^{t}v)}{|x|^{\alpha }}\right) \frac{f(e^{t}v)e^{t}w}{|x|^{\alpha }}\textrm{d}x \\&\ \ \ + \frac{(4-\mu -2\alpha )s}{2e^{(4-\mu -2\alpha )t}}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(e^{t}v)}{|x|^{\alpha }}\right) \frac{F(e^{t}v)}{|x|^{\alpha }}\textrm{d}x \\&\ \ \ -\frac{s}{e^{(4-\mu -2\alpha )t}}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(e^{t}v)}{|x|^{\alpha }}\right) \frac{f(e^{t}v)e^{t}v}{|x|^{\alpha }}\textrm{d}x \\&= \left\langle {\Phi }'(\beta (v,t)),\beta (w,t)\right\rangle +s J(\beta (v,t)). \end{aligned}$$

The norm of the derivative of the ${\mathcal {C}}^1$ restriction functional ${\tilde{\Phi }}|_{{\mathcal {S}}_a\times {{\mathbb {R}}} }$ is defined by

$$\begin{aligned} \Vert {\tilde{\Phi }}|_{{\mathcal {S}}_a\times {{\mathbb {R}}} }'(u,t)\Vert =\sup _{(v,s)\in {\tilde{T}}_{u,t},\Vert (v,s)\Vert _H=1} \left\langle {\tilde{\Phi }}|_{{\mathcal {S}}_a\times {{\mathbb {R}}} }'(u,t),(v,s)\right\rangle . \end{aligned}$$

In the same way as [21, Proposition 2.2], we have the following proposition.

Proposition 2.7

Assume that ${\tilde{\Phi }}$ has a mountain pass geometry on the constraint ${\mathcal {S}}_a\times {{\mathbb {R}}}$. Let $a>0$ and $\{{\tilde{g}}_n\} \subset {\tilde{\Gamma }}_a$ be such that

$$\begin{aligned} \max _{\tau \in [0,1]}{\tilde{\Phi }}({\tilde{g}}_n(\tau ))\le {\tilde{c}}(a)+\frac{1}{n}, \ \ \ \ \forall \ n\in {\mathbb {N}}. \end{aligned}$$

Then there exists a sequence $\{(v_n,t_n)\}\subset {\mathcal {S}}_a\times {{\mathbb {R}}}$ such that

• 1. ${\tilde{\Phi }}(v_n,t_n)\in \left[ {\tilde{c}}(a)-\frac{1}{n},{\tilde{c}}(a)+\frac{1}{n}\right]$;

• 2. $\min _{\tau \in [0,1]}\Vert (v_n,t_n)-{\tilde{g}}_n(\tau )\Vert _H\le \frac{1}{\sqrt{n}}$;

• 3. $\Vert {\tilde{\Phi }}|_{{\mathcal {S}}_a\times {{\mathbb {R}}} }'(v_n,t_n)\Vert \le \frac{2}{\sqrt{n}}$, i.e.,

  $$\begin{aligned} |\langle {\tilde{\Phi }}'(v_n,t_n),(v,s)\rangle |\le \frac{2}{\sqrt{n}}\Vert (v,s)\Vert _{H}, \ \ \ \ \forall \ (v,s)\in {\tilde{T}}_{v_n,t_n}. \end{aligned}$$

Note that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}{\tilde{\Phi }}(v,t)&= \left\langle {\tilde{\Phi }}'(v,t),(0,1)\right\rangle \nonumber \\&=e^{2t}\Vert \nabla v\Vert _2^2+ \frac{(4-\mu -2\alpha )}{2e^{(4-\mu -2\alpha )t}}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(e^{t}v)}{|x|^{\alpha }}\right) \frac{F(e^{t}v)}{|x|^{\alpha }}\textrm{d}x\nonumber \\&\ \ \ \ -\frac{1}{e^{(4-\mu -2\alpha )t}}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(e^{t}v)}{|x|^{\alpha }}\right) \frac{f(e^{t}v)e^{t}v}{|x|^{\alpha }}\textrm{d}x \nonumber \\&= J(\beta (v, t)), \ \ \ \ \forall \ (v,t)\in H. \end{aligned}$$

(2.10)

With the aforementioned lemmas, we can get the desired sequence as follows.

Lemma 2.8

Assume that (F1)–(F3) and (F5) hold. Then there exists a bounded sequence $\{u_n\}\subset {\mathcal {S}}_a$ such that

$$\begin{aligned} \Phi (u_n)\rightarrow {c}(a)>0, \ \ \Phi |_{{\mathcal {S}}_a}'(u_n) \rightarrow 0\ \ \text{ and }\ \ J(u_n)\rightarrow 0. \end{aligned}$$

(2.11)

Proof

Let

$$\begin{aligned} u_n=\beta (v_n, t_n) \ \ \hbox {and}\ \ g_n(\tau )=\beta ( {\tilde{g}}_n(\tau ))\ \hbox {for}\ \tau \in [0,1], \end{aligned}$$

(2.12)

where $\beta$ is defined by (1.6), $v_n, t_n$, and ${\tilde{g}}_n$ are given in Proposition 2.7. Then $u_n\in {\mathcal {S}}_a$ and $g_n\in \Gamma _a$ by (ii) of Lemma 2.3. Moreover, by (2.9), (2.10), Lemma 2.6, and Proposition 2.7, we have

$$\begin{aligned} \Phi (u_n)={\tilde{\Phi }}(v_n,t_n)\in \left[ {c}(a)-\frac{1}{n},{c}(a)+\frac{1}{n}\right] , \end{aligned}$$

(2.13)

and

$$\begin{aligned} J(u_n)= \left\langle {\tilde{\Phi }}'(v_n,t_n),(0,1)\right\rangle \rightarrow 0. \end{aligned}$$

(2.14)

By (F3), we have

$$\begin{aligned} {c}(a)+o(1)&= \Phi (u_n)-\frac{1}{2}J(u_n) \\&= \frac{-6+\mu +2\alpha }{4}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(u)}{|x|^{\alpha }}\right) \frac{F(u)}{|x|^{\alpha }}\textrm{d}x\\ {}&\quad +\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(u)}{|x|^{\alpha }}\right) \frac{f(u)u}{|x|^{\alpha }}\textrm{d}x \\&\ge \frac{6-\mu -2\alpha }{4}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(u)}{|x|^{\alpha }}\right) \frac{F(u)}{|x|^{\alpha }}\textrm{d}x, \end{aligned}$$

which implies that $\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(u)}{|x|^{\alpha }}\right) \frac{F(u)}{|x|^{\alpha }}\textrm{d}x$ is bounded. Using (F3) and (F4), we know that for any $\delta >0$ there exists $R_{\delta }>0$ such that

$$\begin{aligned} f(t)t\ge \delta F(t)>0, \ \ \ \ \forall \ |t|\ge R_{\delta }. \end{aligned}$$

Then we have

$$\begin{aligned}&{c}(a)+o(1) \\&\quad = \Phi (u_n)-\frac{1}{4}J(u_n) \\&\quad =\frac{1}{4}\Vert \nabla u_n\Vert _2^2+\frac{1}{4}\int _{|u_n|<R_{4}}\left( I_{\mu }\times \frac{F(u)}{|x|^{\alpha }}\right) \frac{\left[ f(u)u-\frac{8-\mu -2\alpha }{2}F(u)\right] }{|x|^{\alpha }}\textrm{d}x \\&\qquad \ +\frac{1}{4}\int _{|u_n|\ge R_{4}}\left( I_{\mu }\times \frac{F(u)}{|x|^{\alpha }}\right) \frac{\left[ f(u)u-\frac{8-\mu -2\alpha }{2}F(u)\right] }{|x|^{\alpha }}\textrm{d}x \\&\quad \ge \frac{1}{4}\Vert \nabla u_n\Vert _2^2-\frac{1}{4}\left[ \int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{\left[ \frac{8-\mu -2\alpha }{2}F(u)-f(u)u\right] \chi _{|u_{n}|<R_{4}}}{|x|^{\alpha }}\right) \right. \\&\quad \left. \frac{\left[ \frac{8-\mu -2\alpha }{2}F(u)-f(u)u \right] \chi _{|u_{n}|<R_{4}}}{|x|^{\alpha }}\textrm{d}x\right] ^{\frac{1}{2}} \\&\qquad \times \left[ \int _{{\mathbb {R}}^{2}}\left( I_{\mu }\times \frac{F(u)}{|x|^{\alpha }}\right) \frac{F(u)}{|x|^{\alpha }}\textrm{d}x\right] ^{\frac{1}{2}} \\&\quad \ge \frac{1}{4}\Vert \nabla u_n\Vert _2^2-\frac{C}{4}\left[ \int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{\left[ C|u_{n}|^{\frac{4-\mu -2\alpha }{2}}\right] }{|x|^{\alpha }}\right) \frac{\left[ C|u_{n}|^{\frac{4-\mu -2\alpha }{2}}\right] }{|x|^{\alpha }}\textrm{d}x\right] ^{\frac{1}{2}} \\&\quad \ge \frac{1}{4}\Vert \nabla u_n\Vert _2^2-C\Vert u_{n} \Vert _{2}^{(4-\mu -2\alpha )/2} \quad \mathrm{by \ taking} \ \delta =4, \end{aligned}$$

which implies that $\{u_n\}$ is bounded in $H^1({{\mathbb {R}}} ^2)$. To finish the proof, it remains to prove that $\Phi |_{{\mathcal {S}}_a}'(u_n) \rightarrow 0$, i.e., $\left\langle \Phi '(u_n),w\right\rangle \rightarrow 0$ for all $w\in T_{u_n}$. For this, we just need to show that $\{(\beta (w,-t_n),0)\}\subset T_{v_n,t_n}$ and $\{(\beta (w,-t_n),0)\}$ is bounded in H since

$$\begin{aligned} \left\langle \Phi '(u_n),w\right\rangle =\left\langle {\tilde{\Phi }}'(v_n,t_n),(\beta (w,-t_n),0)\right\rangle \le \frac{2}{\sqrt{n}}\Vert (\beta (w,-t_n),0)\Vert _H, \ \ \ \ \forall \ w\in T_{u_n}. \end{aligned}$$

Indeed, for any $w\in T_{u_n}$, i.e.,

$$\begin{aligned} \int _{{{\mathbb {R}}} ^2}u_nw\textrm{d}x=\int _{{{\mathbb {R}}} ^2}e^{t_n}v_n(e^{t_n}x)w(x)\textrm{d}x=0, \end{aligned}$$

we have

$$\begin{aligned} \int _{{{\mathbb {R}}} ^2}v_n(x)\beta (w,-t_n)(x)\textrm{d}x =\int _{{{\mathbb {R}}} ^2}v_n(x)e^{-t_n}w(e^{-t_n}x)\textrm{d}x =\int _{{{\mathbb {R}}} ^2}e^{t_n}v_n(e^{t_n}x)w(x)\textrm{d}x=0, \end{aligned}$$

which implies

$$\begin{aligned} (\beta (w,-t_n),0)\in T_{v_n,t_n}. \end{aligned}$$

(2.15)

Moreover, by (ii) of Proposition 2.7, we have

$$\begin{aligned} |t_n|\le \min _{\tau \in [0,1]}\Vert (v_n,t_n)-{\tilde{g}}_n(\tau )\Vert _H\le 1\ \text{ for } \text{ large }\ n\in {\mathbb {N}}, \end{aligned}$$

which leads to

$$\begin{aligned}{} & {} \Vert (\beta (w,-t_n),0)\Vert _H^2=\Vert \beta (w,-t_n)\Vert ^2 \\ {}{} & {} \quad = e^{-2t_n}\Vert \nabla w\Vert _2^2+\Vert w\Vert _2^2 \le \ e^{2}\Vert w\Vert ^2\ \text{ for } \text{ large }\ n\in {\mathbb {N}}. \end{aligned}$$

This shows that $\{(\beta (w,-t_n),0)\}\subset T_{v_n,t_n}$ is bounded in H. Jointly with (2.15), we get $\Phi |_{{\mathcal {S}}_a}'(u_n) \rightarrow 0$. From this, (2.13) and (2.14), we conclude that $\{u_n\}$, defined by (2.12), is bounded, and satisfies (2.11). The proof is completed. $\square$

3 Energy Estimates for Minimax Level

In this subsection, we give a precise estimation for the energy level c(a) given by (2.8), which helps us to restore the compactness in the critical exponential case in next subsection.

Let $\kappa :=\liminf _{|t|\rightarrow \infty }\frac{f(t)}{e^{\alpha _0 t^2}}$. By (F4), we know that $\kappa >0$. Then we can choose $d>0$ such that

$$\begin{aligned} \kappa >\frac{(2-\mu )(3-\mu )(4-\mu )[(4-2\alpha -\mu )(1+\varepsilon )\pi ]^{\frac{6-2\alpha -\mu }{2}}}{2e \pi ^2 d^{4-2\alpha -\mu } \alpha _0^{\frac{4-2\alpha -\mu }{2}}}. \end{aligned}$$

(3.1)

For large $n\in {\mathcal {N}}$, let $R_n\ge d$ be such that

$$\begin{aligned} a&= \frac{d^2}{16\log n}\left( 1+2\log 2+2\log ^22-\frac{4}{n^2}-\frac{8}{n^2}\log n\right) \\&\ \ \ \ +\frac{\log ^22}{48(2R_n-d)\log n}\left( 8R_n^3+4R_n^2d-10R_nd^2+3d^3\right) . \end{aligned}$$

Then one has

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{R_n^2}{\log n}= \frac{12a}{\log ^22}. \end{aligned}$$

Now we define the following new Moser-type functions $w_n(x)$ supported in $B_{d}:=B_{d}(0)$

$$\begin{aligned} w_n(x)=\frac{1}{\sqrt{2\pi }} {\left\{ \begin{array}{ll} \sqrt{\log n}, \ \ {} &{} 0\le |x|\le d/n;\\ \frac{\log (d/|x|)}{\sqrt{\log n}}, \ \ {} &{} d/n\le |x|\le d/2;\\ \frac{2(R_n-|x|)\log 2}{(2R_n-d)\sqrt{\log n}}, \ \ {} &{} d/2\le |x|\le R_n;\\ 0, \ \ {} &{} |x|\ge R_n. \end{array}\right. } \end{aligned}$$

(3.2)

Computing directly, we get that for large $n\in {\mathcal {N}}$,

$$\begin{aligned} \Vert \nabla w_n\Vert _2^2=\int _{{{\mathbb {R}}} ^2}|\nabla w_n|^2\textrm{d}x = 1-\frac{\log 2}{\log n} +\frac{(2R_n+d)\log ^22}{2(2R_n-d)\log n}\le 1, \end{aligned}$$

(3.3)

and

$$\begin{aligned} \Vert w_n\Vert _2^2&= \int _{{{\mathbb {R}}} ^2}|w_n|^2\textrm{d}x = \int _{0}^{d/n}(\log n)r\textrm{d}r\\&\quad +\int _{d/n}^{d/2}\frac{\log ^2(d/r)}{\log n}r\textrm{d}r +\int _{d/2}^{R_n}\frac{4(R_n-r)^2\log ^22}{(2R_n-d)^2\log n}r\textrm{d}r \\&= \frac{d^2}{16\log n}\left( 1+2\log 2+2\log ^22-\frac{4}{n^2}-\frac{8}{n^2}\log n\right) \\&\ \ \ \ +\frac{\log ^22}{48(2R_n-d)\log n}\left( 8R_n^3+4R_n^2d-10R_nd^2+3d^3\right) \\&= a. \end{aligned}$$

We also give the observation on the estimation of convolution term.

$$\begin{aligned} \int _{B_{\rho /n}}\int _{B_{\rho /n}}\frac{1}{|x|^\alpha |x-y|^\mu |y|^\alpha }\textrm{d}x\textrm{d}y&\ge \left( \frac{\rho }{n}\right) ^{-2\alpha }\int _{B_{\rho /n}}\int _{B_{\rho /n}}\frac{1}{|x-y|^\mu }\textrm{d}x\textrm{d}y \\&\ge \frac{4\pi ^2}{(2-\mu )(3-\mu )(4-\mu )}\left( \frac{\rho }{n}\right) ^{4-2\alpha -\mu } . \end{aligned}$$

Lemma 3.1

Assume that (F1)–(F4) hold. Then there exists ${\bar{n}}\in {\mathbb {N}}$ such that

$$\begin{aligned} \sup _{t>0}\Phi (t(w_{{\bar{n}}})_{t})< \frac{(4-2\alpha -\mu )\pi }{2\alpha _0}. \end{aligned}$$

(3.4)

Proof

By (F4), we may choose $\varepsilon >0$ small and $t_{\varepsilon }>0$ such that

$$\begin{aligned} f(t)\ge (\kappa -\varepsilon )e^{\alpha _0t^2}, \quad tF(t)\ge \frac{\kappa -\varepsilon }{2\alpha _{0}}e^{\alpha _0t^2}, \ \ \ \ \forall \ |t|\ge t_{\varepsilon }. \end{aligned}$$

(3.5)

Using (3.3), we have

$$\begin{aligned} \Phi (t(w_n)_{t})&= \frac{t^2}{2}\Vert \nabla w_n\Vert _2^2 -\frac{1}{2t^{4-2\alpha -\mu }}\int _{{{{\mathbb {R}}} }^2}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(tw_{n}(y))}{|x-y|^{\mu }|y|^{\alpha }}\right) \frac{F(tw_n(x))}{|x|^{\alpha }} \textrm{d}x \\&\le \frac{t^2}{2}-\frac{1}{2t^{4-2\alpha -\mu }}\int _{{{{\mathbb {R}}} }^2}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(tw_{n}(y))}{|x-y|^{\mu }|y|^{\alpha }}\right) \frac{F(tw_n(x))}{|x|^{\alpha }}\textrm{d}x ,\\&\quad \ \ \ \ \forall \ t > 0, \ \mathrm{for\ large}\ n\in {\mathbb {N}}. \end{aligned}$$

There are three cases to distinguish. Without mentioning, all inequalities hold for large $n\in {\mathbb {N}}$ in the rest of the Lemma.

Case i$t\in \left[ 0,\sqrt{(4-2\alpha -\mu )\pi /2\alpha _0}\right]$. Then by (F3), we have

$$\begin{aligned} \Phi (t(w_n)_{t}){} & {} \le \frac{t^2}{2} -\frac{1}{2t^{4-2\alpha -\mu }}\int _{{{{\mathbb {R}}} }^2}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(tw_{n}(y))}{|x-y|^{\mu }|y|^{\alpha }}\right) \frac{F(tw_n(x))}{|x|^{\alpha }}\textrm{d}x\\{} & {} \le \frac{t^2}{2}\le \frac{(4-2\alpha -\mu )\pi }{4\alpha _0}, \end{aligned}$$

which yields the existence of ${\bar{n}}\in {\mathbb {N}}$ satisfying (3.4).

Case ii $t\in \left[ \sqrt{(4-2\beta -\mu )\pi /2\alpha _0},\sqrt{(4-2\alpha -\mu )(1+\varepsilon )\pi /\alpha _0}\right]$. In this case, $tw_n(x)\ge t_{\varepsilon }$ for $x\in B_{d/n}$ and $n\in {\mathbb {N}}$ large. Then it follows from (3.2) and (3.5) that

$$\begin{aligned}&\ \frac{1}{2t^{4-2\alpha -\mu }}\int _{{{{\mathbb {R}}} }^2}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(tw_{n}(y))}{|x-y|^{\mu }|y|^{\alpha }}\right) \frac{F(tw_n(x))}{|x|^{\alpha }}\textrm{d}x \\&\quad \ge \ \frac{1}{2t^{4-2\alpha -\mu }}\int _{B_{d/n}}\left( \int _{B_{d/n}}\frac{F(tw_{n}(y))}{|x-y|^{\mu }|y|^{\alpha }}\right) \frac{F(tw_n(x))}{|x|^{\alpha }}\textrm{d}x \\&\quad \ge \frac{\pi ^3 d^{4-2\alpha -\mu }(\kappa -\varepsilon )^{2}e^{\alpha _{0} \pi ^{-1}t^{2}\log n }}{(2-\mu )(3-\mu )(4-\mu )n^{4-2\alpha -\mu } \alpha _{0}^{2}\log n t^{6-2\alpha -\mu }} , \end{aligned}$$

then

$$\begin{aligned} \Phi (t(w_n)_{t})&\le \frac{t^2}{2}-\frac{1}{2t^{4-2\alpha -\mu }}\int _{{{{\mathbb {R}}} }^2}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(tw_{n}(y))}{|x-y|^{\mu }|y|^{\alpha }}\textrm{d}y\right) \frac{F(tw_n(x))}{|x|^{\alpha }}\textrm{d}x \\&\le \frac{t^2}{2}- \frac{\pi ^3 d^{4-2\alpha -\mu }(\kappa -\varepsilon )^{2}e^{\alpha _{0} \pi ^{-1}t^{2}\log n }}{(2-\mu )(3-\mu )(4-\mu )n^{4-2\alpha -\mu } \alpha _{0}^{2}\log n t^{6-2\alpha -\mu }} \\&\le \frac{t^2}{2}-\frac{\pi ^3 d^{4-2\alpha -\mu } \alpha _0^{\frac{2-2\alpha -\mu }{2}}}{(2-\mu )(3-\mu )(4-\mu )[(4-2\alpha -\mu )(1+\varepsilon )\pi ]^{\frac{6-2\alpha -\mu }{2}}}\\&\quad \frac{(\kappa -\varepsilon )^{2}e^{\alpha _{0} \pi ^{-1}t^{2}\log n }}{n^{4-2\alpha -\mu } \log n } \\&:= \varphi _n(t). \end{aligned}$$

Choosing $t_n>0$ be such that $\varphi _n'(t_n)=0$, then we have

$$\begin{aligned} 1=\frac{2 \pi ^2 d^{4-2\alpha -\mu } \alpha _0^{\frac{4-2\alpha -\mu }{2}}}{(2-\mu )(3-\mu )(4-\mu )[(4-2\alpha -\mu )(1+\varepsilon )\pi ]^{\frac{6-2\alpha -\mu }{2}}} \frac{(\kappa -\varepsilon )^{2}e^{\alpha _{0} \pi ^{-1}t_n^{2}\log n }}{n^{4-2\alpha -\mu }}. \end{aligned}$$

Let

$$\begin{aligned}{} & {} B_1=2 \pi ^2 d^{4-2\alpha -\mu } \alpha _0^{\frac{4-2\alpha -\mu }{2}}, \ \\{} & {} \qquad B_2=(2-\mu )(3-\mu )(4-\mu )[(4-2\alpha -\mu )(1+\varepsilon )\pi ]^{\frac{6-2\alpha -\mu }{2}}, \end{aligned}$$

then it follows that

$$\begin{aligned} t_n^2&= \frac{(4-2\alpha -\mu )\pi }{\alpha _0}\left[ 1+\frac{\log B_2-\log [B_1(\kappa -\varepsilon )^{2}]}{(4-2\alpha -\mu )\log n}\right] \nonumber \\&= \frac{(4-2\alpha -\mu )\pi }{\alpha _0}-\frac{\pi }{\alpha _0\log n}\log \frac{[B_1(\kappa -\varepsilon )^{2}]}{B_2}, \end{aligned}$$

(3.6)

and

$$\begin{aligned} \varphi _n(t)\le \varphi _n(t_n)=\frac{t_n^2}{2}-\frac{\pi }{2\alpha _0\log n}, \ \ \ \ \forall \ t\ge 0. \end{aligned}$$

(3.7)

Using (3.6) and (3.7), we are led to

$$\begin{aligned} \varphi _n(t)&\le \frac{t_n^2}{2}-\frac{\pi }{2\alpha _0\log n}\\&= \frac{(4-2\alpha -\mu )\pi }{2\alpha _0}-\frac{\pi }{2\alpha _0\log n}\log \frac{B_1(\kappa -\varepsilon )^{2}}{B_2}-\frac{\pi }{2\alpha _0\log n}\\&= \frac{(4-2\alpha -\mu )\pi }{2\alpha _0}-\frac{\pi }{2\alpha _0\log n}\left[ 1+\log \frac{B_1(\kappa -\varepsilon )^{2}}{B_2}\right] , \end{aligned}$$

where from (3.1), we know that

$$\begin{aligned} 1+\log \frac{2 \pi ^2 d^{4-2\alpha -\mu } \alpha _0^{\frac{4-2\alpha -\mu }{2}}(\kappa -\varepsilon )^{2}}{(2-\mu )(3-\mu )(4-\mu )[(4-2\alpha -\mu )(1+\varepsilon )\pi ]^{\frac{6-2\alpha -\mu }{2}}} > 0, \end{aligned}$$

thus we have

$$\begin{aligned} \Phi (t(w_n)_{t}) \le \frac{(4-2\alpha -\mu )\pi }{2\alpha _0}-\frac{\pi }{2\alpha _0\log n}\left[ 1+\log \frac{B_1(\kappa -\varepsilon )^{2}}{B_2}\right] < \frac{(4-2\alpha -\mu )\pi }{2\alpha _0}. \end{aligned}$$

Then we deduce that (3.4) holds for some ${\bar{n}}\in {\mathbb {N}}$.

Case iv$t\in (\sqrt{(4-2\beta -\mu )(1+\varepsilon )\pi /\alpha _0},\infty )$. Since $tw_n(x)\ge t_{\varepsilon }$ for $x\in B_{d/n}$ and ${\bar{n}}\in {\mathbb {N}}$ large, we deduce from (3.2) that

$$\begin{aligned} \Phi (t(w_n)_{t})&\le \frac{t^2}{2}-\frac{1}{2t^{4-2\alpha -\mu }}\int _{{{{\mathbb {R}}} }^2}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(tw_{n}(y))}{|x-y|^{\mu }|y|^{\alpha }}\textrm{d}y\right) \frac{F(tw_n(x))}{|x|^{\alpha }}\textrm{d}x \\&\le \frac{t^2}{2}- \frac{\pi ^3 d^{4-2\alpha -\mu }}{(2-\mu )(3-\mu )(4-\mu )} \frac{(\kappa -\varepsilon )^{2}e^{\alpha _{0} \pi ^{-1}t^{2}\log n }}{n^{4-2\alpha -\mu } \alpha _{0}^{2}\log n t^{6-2\alpha -\mu }} \\&\le \frac{(4-2\beta -\mu )(1+\varepsilon )\pi }{2\alpha _0} \\&\ \ \ -\frac{\pi ^3 d^{4-2\alpha -\mu }}{(2-\mu )(3-\mu )(4-\mu )} \frac{(\kappa -\varepsilon )^{2}e^{(4-2\beta -\mu )\varepsilon \log n }}{\alpha _{0}^{2}\log n [(4-2\beta -\mu )(1+\varepsilon )\pi /\alpha _0]^{\frac{6-2\alpha -\mu }{2}}} \\&\le \frac{(4-2\beta -\mu )(1+\varepsilon )\pi }{3\alpha _0}, \end{aligned}$$

where we have used the fact that the function

$$\begin{aligned} \varphi _n(t):= \frac{t^2}{2}- \frac{\pi ^3 d^{4-2\alpha -\mu }}{(2-\mu )(3-\mu )(4-\mu )} \frac{(\kappa -\varepsilon )^{2}e^{\alpha _{0} \pi ^{-1}t^{2}\log n }}{n^{4-2\alpha -\mu } \alpha _{0}^{2}\log n t^{6-2\alpha -\mu }}, \end{aligned}$$

is decreasing on $t\in \left( \sqrt{\frac{(4-2\alpha -\mu )\pi }{\alpha _0}(1+\varepsilon )}, +\infty \right)$ for large n. In fact,

$$\begin{aligned}{} & {} \varphi _n'(t)=t-\frac{\pi ^3 d^{4-2\alpha -\mu }}{(2-\mu )(3-\mu )(4-\mu )} \frac{(\kappa -\varepsilon )^{2}}{n^{4-2\alpha -\mu } \alpha _{0}^{2}\log n }\cdot \frac{e^{\alpha _{0} \pi ^{-1}t^{2}\log n }}{t^{7-2\alpha -\mu }}\\{} & {} \quad \left( \frac{2\alpha _0 \log n t^2}{\pi }-(6-2\alpha -\mu )\right) . \end{aligned}$$

Assume that $s_n\ge \sqrt{\frac{(4-2\alpha -\mu )\pi }{\alpha _0}(1+\varepsilon )}$ such that $\varphi _n'(s_n)=0$ for large n. Then

$$\begin{aligned}{} & {} s_n^{8-2\alpha -\mu } =\frac{\pi ^2 d^{4-2\alpha -\mu }}{(2-\mu )(3-\mu )(4-\mu )} \frac{(\kappa -\varepsilon )^{2}}{n^{4-2\alpha -\mu } \alpha _{0}^{2} }\cdot \\{} & {} \quad \left( 2\alpha _0 s_n^2-\frac{(6-2\alpha -\mu )\pi }{\log n}\right) e^{\alpha _{0} \pi ^{-1}s_n^{2}\log n }, \end{aligned}$$

which yields

$$\begin{aligned} s_n^2&= \frac{(4-2\alpha -\mu )\pi }{\alpha _0}\left[ 1+\right. \\&\left. \ \ \ \ \frac{\log [(2-\mu )(3-\mu )(4-\mu )\alpha _0^2 s_n^{8-2\alpha -\mu }] -\log \left( \pi ^2 d^{4-2\alpha -\mu }(\kappa -\varepsilon )^2 \left( 2\alpha _0 s_n^2-\frac{(6-2\alpha -\mu )\pi }{\log n}\right) \right) }{(4-2\alpha -\mu )\log n}\right] \\&= \frac{(4-2\alpha -\mu )\pi }{\alpha _0}+\frac{\pi }{\alpha _0\log n} \log \frac{[(2-\mu )(3-\mu )(4-\mu )\alpha _0^2 s_n^{8-2\alpha -\mu }]}{\pi ^2 d^{4-2\alpha -\mu }(\kappa -\varepsilon )^2 \left( 2\alpha _0 s_n^2-\frac{(6-2\alpha -\mu )\pi }{\log n}\right) }. \end{aligned}$$

This implies that $\lim _{n\rightarrow \infty }s_n^2=\frac{(4-2\alpha -\mu )\pi }{\alpha _0}$, a contradiction. So $\varphi _n(t)$ is decreasing for large n when $t\in \left( \sqrt{\frac{(4-2\alpha -\mu )\pi }{\alpha _0}(1+\varepsilon )}, +\infty \right)$. Thus (3.4) holds for some ${\bar{n}}\in {\mathbb {N}}$. Till now, we have completed the proof. $\square$

Lemma 3.2

Assume that (F1)–(F5) hold. Then ${c}(a)<2\pi /\alpha _0$ for any $a>0$.

Proof

Let $w_{{\bar{n}}}$ be given by Lemma 3.1. Since $\Vert \nabla t(w_{{\bar{n}}})_{t}\Vert _2^2=t^2\Vert \nabla w_{{\bar{n}}}\Vert _2^2$, we know that there exists $t_w>0$ small enough and $T_w>0$ large enough such that $\Vert \nabla t_w(w_{{\bar{n}}})_{t_w}\Vert _2^2\le K(a)$ and $\Phi (T_w(w_{{\bar{n}}})_{T_w})<0$ by (2.7). Set

$$\begin{aligned} g_0(\tau )=[(1-\tau ) t_w+\tau T_w](w_{{\bar{n}}})_{(1-\tau ) t_w+\tau T_w}, \ \ \ \ \forall \ \tau \in [0,1]. \end{aligned}$$

Then $g_0\in \Gamma _a$. Jointly with the definition of c(a), we have ${c}(a)<2\pi /\alpha _0$ for any $a>0$. $\square$

Lemma 3.3

Assume that f satisfies (F1)–(F3) and (F4$'$) with $\gamma >\gamma ^*(a)$. Then ${c}(a)<[(4-2\mu -\beta )\pi ]/(2\alpha _0)$, where $\gamma ^*(a)$ is given by (1.4).

Proof

Since

$$\begin{aligned} {\mathcal {C}}_{\frac{4p}{4-2\alpha -\mu }}^{-\frac{4-2\alpha -\mu }{2}}=\inf _{u\in H^1({{\mathbb {R}}} ^2)\setminus \{0\}} \frac{\Vert \nabla u\Vert _2^{2p-(4-2\alpha -\mu )}\Vert u\Vert _2^{4-2\alpha -\mu }}{\Vert u\Vert _{\frac{4p}{4-2\alpha -\mu }}^{2p}}, \end{aligned}$$

we can choose $v_n\in {\mathcal {S}}_a$ such that

$$\begin{aligned} {\mathcal {C}}_{\frac{4p}{4-2\alpha -\mu }}^{-\frac{4-2\alpha -\mu }{2}}\le \frac{\Vert \nabla v_{n}\Vert _2^{2p-(4-2\alpha -\mu )}a^{\frac{4-2\alpha -\mu }{2}}}{\Vert u\Vert _{\frac{4p}{4-2\alpha -\mu }}^{2p}}< {\mathcal {C}}_{\frac{4p}{4-2\alpha -\mu }}^{-\frac{4-2\alpha -\mu }{2}}+\frac{1}{n}, \ \ \ \ \forall \ n\in {\mathbb {N}}. \end{aligned}$$

(3.8)

Note that

$$\begin{aligned} \Phi (t(v_n)_{t})&= \frac{t^2}{2}\Vert \nabla v_n\Vert _2^2 -\frac{1}{2t^{4-2\alpha -\mu }}\int _{{{{\mathbb {R}}} }^2}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(tv_{n}(y))}{|x-y|^{\mu }|y|^{\alpha }}\right) \frac{F(tv_n(x))}{|x|^{\alpha }}\textrm{d}x \nonumber \\&\le \frac{t^2}{2}\Vert \nabla v_n\Vert _2^2 -\frac{1}{2t^{4-2\alpha -\mu }}\int _{{{{\mathbb {R}}} }^2}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{\gamma |tv_{n}(y)|^{p}}{|x-y|^{\mu }|y|^{\alpha }}\right) \frac{\gamma |tv_n(x)|^{p}}{|x|^{\alpha }}\textrm{d}x \nonumber \\&\le \frac{t^2}{2}\Vert \nabla v_n\Vert _2^2-\frac{\gamma ^2 {\mathcal {C}}(2,\mu ,\alpha ) t^{2p-(4-2\alpha -\mu )}}{2}\Vert v_n\Vert _{\frac{4p}{4-2\alpha -\mu }}^{2p}\nonumber \\&:= g_n(t) , \ \ \ \ \forall \ t > 0, \ n\in {\mathbb {N}}. \end{aligned}$$

(3.9)

Let $g_n'(t_n)=0$, then one has

$$\begin{aligned} t_n^{2p-(6-2\alpha -\mu )}=\frac{2\Vert \nabla v_n\Vert _2^2}{[2p-(4-2\alpha -\mu )]\gamma ^2 {\mathcal {C}}(2,\mu ,\alpha )\Vert v_n\Vert _{\frac{4p}{4-2\alpha -\mu }}^{2p}}. \end{aligned}$$

It is easy to see that $g_n(t)\le g_n(t_n)$ for all $t>0$. We define that

$$\begin{aligned} A:=\frac{2^{\frac{(8-2\alpha -\mu )-2p}{2p-(6-2\alpha -\mu )}}[2p-(6-2\alpha -\mu )]}{[2p-(4-2\alpha -\mu )]^{\frac{2p-(4-2\alpha -\mu )}{2p-(6-2\alpha -\mu )}}}. \end{aligned}$$

(3.10)

Then it follows from (3.8) and (3.9) that

$$\begin{aligned} \Phi (t(v_n)_{t})&\le g_n(t_n) =A\left( \frac{1}{\gamma ^{2}{\mathcal {C}}(2,\mu ,\alpha )}\right) ^{\frac{2}{2p-(6-2\alpha -\mu )}}\nonumber \\&\left( \frac{\Vert \nabla v_{n}\Vert _{2}^{2p-(4-2\alpha -\mu )}}{\Vert v_{n}\Vert _{\frac{4p}{4-2\alpha -\mu }}^{2p}}\right) ^{\frac{2}{2p-(6-2\alpha -\mu )}} \nonumber \\&\le A\left( \frac{1}{\gamma ^{2}{\mathcal {C}}(2,\mu ,\alpha )}\right) ^{\frac{2}{2p-(6-2\alpha -\mu )}} \left( \frac{{\mathcal {C}}_{\frac{4p}{4-2\alpha -\mu }}^{-\frac{4-2\alpha -\mu }{2}}+ \frac{1}{n}}{a^{\frac{4-2\alpha -\mu }{2}}}\right) ^{\frac{2}{2p-(6-2\alpha -\mu )}} , \nonumber \\&\ \ \ \ \forall \ t > 0, \ n\in {\mathbb {N}}. \end{aligned}$$

(3.11)

Since $p>\frac{6-2\alpha -\mu }{2}$ and $\gamma >\gamma ^*(a)$, then there exists $\epsilon _0>0$ such that

$$\begin{aligned} \gamma&=\gamma ^*(a) (1-\epsilon _0)^{[(6-2\alpha -\mu )-2p]/4}\\&= \frac{A^{\frac{2p-(6-2\alpha -\mu )}{4}}}{ \sqrt{{\mathcal {C}}(2,\mu ,\alpha )}a^{\frac{4-2\alpha -\mu }{4}}C_{ \frac{4p}{4-2\alpha -\mu }}^{\frac{4-2\alpha -\mu }{4}}}\left[ \frac{2\alpha _{0}}{(4-2\alpha -\mu )\pi (1-\varepsilon _{0})}\right] ^{\frac{2p-(6-2\alpha -\mu )}{4}}, \end{aligned}$$

which together with (3.11) imply that

$$\begin{aligned} \Phi (t(v_n)_{t}) \le \left( \frac{ {\mathcal {C}}_{\frac{4p}{4-2\alpha -\mu }}^{-\frac{4-2\alpha -\mu }{2}}+\frac{1}{n}}{ {\mathcal {C}}_{\frac{4p}{4-2\alpha -\mu }}^{-\frac{4-2\alpha -\mu }{2}}}\right) ^{\frac{2}{2p-(6-2\alpha -\mu )}}\frac{(4-2\alpha -\mu )\pi (1-\epsilon _0)}{2\alpha _0}, \ \ \ \ \forall \ t > 0, \ n\in {\mathbb {N}}, \end{aligned}$$

which implies that there exists ${\bar{n}}\in {\mathbb {N}}$ large enough such that

$$\begin{aligned} \max _{t>0}\Phi (t(v_{{\bar{n}}})_{t})<\frac{(4-2\alpha -\mu )\pi }{2\alpha _0}. \end{aligned}$$

(3.12)

Replacing $w_{{\bar{n}}}$ by $v_{{\bar{n}}}$ in the proof of Lemma 3.2, we can get ${c}(a)\le \max _{t>0}\Phi (t(v_{{\bar{n}}})_{t})$ for any $\gamma >\gamma ^*(a)$. From this and (3.12), we derived the desired conclusion, and so the proof is completed. $\square$

4 Restore the Compactness

Let us first establish the following two convergence results which contribute to the final proof.

Lemma 4.1

Assume that $\alpha <\mu$, $u_{n}\rightharpoonup u$ in $H_r^1({{\mathbb {R}}} ^2)$ and

$$\begin{aligned} \int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{f(u_{n}(x))u_{n}(x)}{|x|^{\alpha }}\textrm{d}x\le {\mathcal {K}}, \end{aligned}$$

for some constant ${\mathcal {K}}>0$. Then for every $\varphi \in {\mathcal {C}}_{0}^{\infty }({{\mathbb {R}}} ^{2})$, we have

$$\begin{aligned}{} & {} \lim _{n\rightarrow \infty }\int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{f(u_{n}(x))\varphi (x)}{|x|^{\alpha }} \textrm{d}x\\{} & {} \quad =\int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{f(u(x))\varphi (x)}{|x|^{\alpha }}\textrm{d}x. \end{aligned}$$

Proof

By the Fatou’s Lemma we have

$$\begin{aligned} \int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{f(u(x))u(x)}{|x|^{\alpha }}\textrm{d}x\le {\mathcal {K}}. \end{aligned}$$

Take $\Omega =\textrm{supp}\ \varphi$, for any given $\varepsilon >0$, let $M_{\varepsilon }:={\mathcal {K}}\Vert \varphi \Vert _{\infty }\varepsilon ^{-1}$, then it follows that for n large enough,

$$\begin{aligned}&\ \int _{(|u_{n}|\ge M_{\varepsilon })\cup (|u|=M_{\varepsilon })}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{|f(u_{n}(x))\varphi (x)|}{|x|^{\alpha }}\textrm{d}x \\ \le&\ \frac{2\varepsilon }{{\mathcal {K}}} \int _{|u_{n}|\ge \frac{M_{\varepsilon }}{2}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{f(u_{n}(x))u_{n}(x)}{|x|^{\alpha }}\textrm{d}x\le 2\varepsilon , \end{aligned}$$

and similarly

$$\begin{aligned}&\int _{|u|\ge M_{\varepsilon }}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{|f(u(x))\varphi (x)|}{|x|^{\alpha }}\textrm{d}x \le \varepsilon . \end{aligned}$$

Since $|f(u_{n})|\chi _{|u_{n}|\le M_{\varepsilon }}\rightarrow |f(u)|\chi _{|u|\le M_{\varepsilon }}$ a.e. in $\Omega {\setminus } D_{\varepsilon }$, where $D_{\varepsilon }=\{x\in \Omega :|u(x)|=M_{\varepsilon }\}$, and

$$\begin{aligned} |f(u_{n})|\chi _{|u_{n}|\le M_{\varepsilon }}\le \max _{|t|\le M_{\varepsilon }}|f(t)|<\infty , \ \ \forall \ x\in \Omega , \end{aligned}$$

the Lebesgue dominated convergence theorem leads to

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{(\Omega \setminus D_{\varepsilon })\cup \{|u_{n}| \le M_{\varepsilon }\}}|f(u_{n})|^{\frac{4}{4-2\alpha -\mu }}\textrm{d}x =\int _{(\Omega \setminus D_{\varepsilon })\cup \{|u|\le M_{\varepsilon }\}}|f(u)|^{\frac{4}{4-2\alpha -\mu }}\textrm{d}x. \end{aligned}$$

Here, we choose $K_{\varepsilon }>t_{0}$ such that

$$\begin{aligned} \Vert \varphi \Vert _{\infty }\left( \frac{M_{0}{\mathcal {K}}}{K_{\varepsilon }}\right) ^{\frac{1}{2}}\left[ 2C(2,\mu ,\alpha )\int _{\Omega } |f(u)|^{\frac{4}{4-2\alpha -\mu }}\textrm{d}x\right] ^{\frac{4-2\alpha -\mu }{4}}<\varepsilon , \end{aligned}$$

and

$$\begin{aligned} \int _{|u|\le M_{\varepsilon }}\left[ \frac{F(u(y))\chi _{|u|\ge K_{\varepsilon }}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right] \frac{|f(u(x))\varphi |}{|x|^{\alpha }}\textrm{d}x<\varepsilon . \end{aligned}$$

With the help of Lemma 2.2, we have

$$\begin{aligned}&\int _{(|u_{n}|\le M_{\varepsilon })\cap (|u|\ne M_{\varepsilon })}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))\chi _{|u_{n}|\ge K_{\varepsilon }}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{|f(u_{n}(x))\varphi |}{|x|^{\alpha }}\textrm{d}x \\&\quad \quad \le \Vert \varphi \Vert _{\infty }\left[ \int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}} \frac{F(u_{n}(y))\chi _{|u_{n}|\ge K_{\varepsilon }}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u_{n}(x))\chi _{|u_{n}|\ge K_{\varepsilon }}}{|x|^{\alpha }}\textrm{d}x\right] ^{\frac{1}{2}} \\&\qquad \times \left[ \int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}} \frac{|f(u_{n}(y))|\chi _{(\Omega \setminus D_{\varepsilon })\cap \{|u_{n}|\le M_{\varepsilon }\}}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{|f(u_{n}(x))|\chi _{(\Omega \setminus D_{\varepsilon })\cap \{|u_{n}|\le M_{\varepsilon }\}}}{|x|^{\alpha }}\textrm{d}x\right] ^{\frac{1}{2}}, \end{aligned}$$

then from (F5) and Proposition 1.2, one has

$$\begin{aligned}&\int _{(|u_{n}|\le M_{\varepsilon })\cap (|u|\ne M_{\varepsilon })}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))\chi _{|u_{n}|\ge K_{\varepsilon }}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{|f(u_{n}(x))\varphi |}{|x|^{\alpha }}\textrm{d}x \\&\quad \le \Vert \varphi \Vert _{\infty } \left[ \int _{|u_{n}|\ge K_{\varepsilon }}\left( \int _{{{\mathbb {R}}} ^{2}} \frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u_{n}(x))}{|x|^{\alpha }}\textrm{d}x\right] ^{\frac{1}{2}} \\&\qquad \times \left[ \int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}} \frac{|f(u_{n}(y))|\chi _{(\Omega \setminus D_{\varepsilon })\cap \{|u_{n}|\le M_{\varepsilon }\}}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{|f(u_{n}(x))|\chi _{(\Omega \setminus D_{\varepsilon })\cap \{|u_{n}|\le M_{\varepsilon }\}}}{|x|^{\alpha }}\textrm{d}x\right] ^{\frac{1}{2}} \\&\quad \le \Vert \varphi \Vert _{\infty } \left[ \int _{|u_{n}|\ge K_{\varepsilon }}\left( \int _{{{\mathbb {R}}} ^{2}} \frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u_{n}(x))}{|x|^{\alpha }}\textrm{d}x\right] ^{\frac{1}{2}} \\&\qquad \times \left[ C(2,\mu ,\alpha ) \int _{(\Omega \setminus D_{\varepsilon })\cap \{|u_{n}|\le M_{\varepsilon }\}}|f(u_{n})|^{\frac{4}{4-2\alpha -\mu }}\textrm{d}x \right] ^{\frac{4-2\alpha -\mu }{4}} \\&\quad \le \Vert \varphi \Vert _{\infty }\left[ \frac{M_{0}}{K_{\varepsilon }}\ \int _{|u_{n}|\ge K_{\varepsilon }}\left( \int _{{{\mathbb {R}}} ^{2}} \frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{f(u_{n}(x))u_{n}(x)}{|x|^{\alpha }}\textrm{d}x\right] ^{\frac{1}{2}} \\&\qquad \times \left[ 2C(2,\mu ,\alpha )\int _{\Omega }|f(u)|^{\frac{4}{4-2\alpha -\mu }}\textrm{d}x +o(1)\right] ^{\frac{4-2\alpha -\mu }{4}} \\&\quad \le \Vert \varphi \Vert _{\infty }\left( \frac{M_{0}{\mathcal {K}}}{K_{\varepsilon }}\right) ^ {\frac{1}{2}}\left[ 2C(2,\mu ,\alpha )\int _{\Omega }|f(u)|^{\frac{4}{4-2\alpha -\mu }}\textrm{d}x\right] ^{\frac{4-2\alpha -\mu }{4}}+o(1)<\varepsilon +o(1) . \end{aligned}$$

For any $x\in {{\mathbb {R}}} ^{2}$, define $\zeta _{n}(x)$ and ${\bar{\zeta }}(x)$ as follows:

$$\begin{aligned} \zeta _{n}(x):= \int _{{{\mathbb {R}}} ^{2}}\frac{|F(u_{n}(y))|\chi _{|u_{n}|\le K_{\varepsilon }}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y, \end{aligned}$$

and

$$\begin{aligned} {\bar{\zeta }}(x):= \int _{{{\mathbb {R}}} ^{2}}\frac{|F(u(y))|\chi _{|u|\le K_{\varepsilon }}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y. \end{aligned}$$

Let us first point out some relationships here. For fixed $x\in {{\mathbb {R}}} ^{2}$, we consider the term

$$\begin{aligned} \int _{|x-y|\le R}\frac{1}{|y|^{\alpha p_{1}}|x-y|^{\mu p_{1}}}\textrm{d}y. \end{aligned}$$

When $x\in {{\mathbb {R}}} ^{2}/B_{2R}(0)$, $y\in B_{R}(x)$, thus $|x-y|<|y|$. Select $p_{1}$ such that $(\mu +\alpha )p_{1}<2$, and thus we have,

$$\begin{aligned} \int _{|x-y|\le R}\frac{1}{|y|^{\alpha p_{1}}|x-y|^{\mu p_{1}}}\textrm{d}y\le \int _{|x-y|\le R}\frac{1}{|x-y|^{(\mu +\alpha )p_{1}}}\textrm{d}y={\mathcal {O}}\left( R^{2-(\mu +\alpha )p_{1}}\right) . \end{aligned}$$

When $x\in B_{2R}(0)$, one has

$$\begin{aligned}&\int _{|x-y|\le R}\frac{1}{|y|^{\alpha p_{1}}|x-y|^{\mu p_{1}}}\textrm{d}y\le \int _{|y|\le R}\frac{1}{|y|^{(\mu +\alpha )p_{1}}}\textrm{d}y\\&\quad +\int _{|x-y|\le 3R}\frac{1}{|x-y|^{(\alpha +\mu ) p_{1}}}\textrm{d}y={\mathcal {O}}\left( R^{2-(\mu +\alpha )p_{1}}\right) . \end{aligned}$$

That is

$$\begin{aligned} \int _{|x-y|\le R}\frac{1}{|y|^{\alpha p_{1}}|x-y|^{\mu p_{1}}}\textrm{d}y\le {\mathcal {O}}\left( R^{2-(\mu +\alpha )p_{1}}\right) . \end{aligned}$$

Choosing q such that $\alpha q<2<\mu q$, one has

$$\begin{aligned}&\int _{|x-y|\ge R}\frac{1}{|y|^{\alpha q}|x-y|^{\mu q}}\textrm{d}y\\&\quad = \int _{({{\mathbb {R}}} ^{2}\setminus B_{R}(x))\cap B_{R}(0)}\frac{1}{|y|^{\alpha q}|x-y|^{\mu q}}\textrm{d}y+\int _{({{\mathbb {R}}} ^{2}\setminus B_{R}(x))\cap ({{\mathbb {R}}} ^{2}\setminus B_{R}(0))} \frac{1}{|y|^{\alpha q}|x-y|^{\mu q}}\textrm{d}y\\&\quad \le \frac{1}{R^{\mu q}}\int _{|y|\le R}\frac{1}{|y|^{\alpha q}}\textrm{d}y+\frac{1}{R^{\alpha q}}\int _{{{\mathbb {R}}} ^{2}\setminus B_{R}(x)}\frac{1}{|x-y|^{\mu q}} \textrm{d}y={\mathcal {O}}\left( R^{2-(\alpha +\mu )q}\right) . \end{aligned}$$

Then from (2.4), we have

$$\begin{aligned} |\zeta _{n}(x)-{\bar{\zeta }}(x)|&\le \int _{{{\mathbb {R}}} ^2}\frac{\left| |F(u_{n}(y))|\chi _{|u_{n}|\le K_{\varepsilon }}-|F(u(y))| \chi _{|u|\le K_{\varepsilon }}\right| }{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \\&\le \left[ \int _{|x-y|\le R}\left| |F(u_{n})|\chi _{|u_{n}|\le K_{\varepsilon }}-|F({\bar{u}})|\chi _{|{\bar{u}}|\le K_{\varepsilon }}\right| ^{p_{1}'}\textrm{d}y\right] ^{\frac{1}{p_{1}'}} \\&\quad \times \left( \int _{|x-y|\le R}\frac{1}{|y|^{\alpha p_{1}}|x-y|^{\mu p_{1}}}\textrm{d}y\right) ^{\frac{1}{p_{1}}} \\&\quad + \left[ \int _{|x-y|>R}\left| |F(u_{n})|\chi _{|u_{n}|\le K_{\varepsilon }}-|F({\bar{u}})|\chi _{|{\bar{u}}|\le K_{\varepsilon }}\right| ^{q'}\textrm{d}y\right] ^{\frac{1}{q'}} \\&\quad \times \left( \int _{|x-y|>R}\frac{1}{|y|^{\alpha q}|x-y|^{\mu q}}\textrm{d}y\right) ^{\frac{1}{q}} \\&\le {\mathcal {O}}\left( R^{2/p_{1}-\alpha -\mu }\right) \left[ \int _{|x-y|\le R}\left| |F(u_{n})| \chi _{|u_{n}|\le K_{\varepsilon }}-|F({\bar{u}})|\chi _{|{\bar{u}}|\le K_{\varepsilon }}\right| ^{p_{1}'}\textrm{d}y\right] ^{\frac{1}{p_{1}'}} \\&\quad + {\mathcal {O}}\left( R^{2/q-\alpha -\mu }\right) \left( \int _{|x-y|>R}\left| |F(u_{n})|\chi _{|u_{n}|\le K_{\varepsilon }}-|F({\bar{u}})|\chi _{|{\bar{u}}|\le K_{\varepsilon }}\right| ^{q'}\textrm{d}y\right) ^{\frac{1}{q'}} \\&\le {\mathcal {O}}\left( R^{2/p_{1}-\alpha -\mu }\right) \left[ \int _{|x-y|\le R}\left| |F(u_{n})| \chi _{|u_{n}|\le K_{\varepsilon }}-|F({\bar{u}})|\chi _{|{\bar{u}}|\le K_{\varepsilon }}\right| ^{p_{1}'}\textrm{d}y\right] ^{\frac{1}{p_{1}'}} \\&\quad + {\mathcal {O}}\left( R^{2/q-\alpha -\mu }\right) \left[ \Vert u_{n}\Vert _{\frac{(4-2\alpha -\mu )pq'}{4}}^{\frac{(4-2\alpha -\mu )p}{4}}+\Vert {\bar{u}}\Vert _{\frac{(4-2\alpha -\mu )pq'}{4}}^{\frac{(4-2\alpha -\mu )p}{4}}\right] \\&\le {\mathcal {O}}\left( R^{2/p_{1}-\alpha -\mu }\right) o_{n}(1)+ {\mathcal {O}}\left( R^{2/q-\alpha -\mu }\right) , \ \ \forall \ \ x\in {{\mathbb {R}}} ^{2}, \end{aligned}$$

which implies that for any $x\in {{\mathbb {R}}} ^{2}$, we have $\zeta _{n}(x)\rightarrow {\bar{\zeta }}(x)$. Similarly, by choosing suitable $p_2$ and $p_3$, then for any $x\in {{\mathbb {R}}} ^{2}$, we know that

$$\begin{aligned}&\ |\zeta _{n}(x)|\le \int _{{{\mathbb {R}}} ^{2}}\frac{|F(u_{n}(x))|\chi _{|u_{n}|\le K_{\varepsilon }}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\\&\quad \le \ \left[ \int _{|x-y|\le R}\left| F(u_{n}(x))\chi _{|u_{n}|\le K_{\varepsilon }} \right| ^{p_{2}'}\textrm{d}y \right] ^{\frac{1}{p_{2}'}}\left[ \int _{|x-y|\le R}\frac{1}{|y|^{\alpha p_{2}}|x-y|^{\mu p_{2}}}\textrm{d}y \right] ^{\frac{1}{p_{2}}}\\&\qquad + \left[ \int _{|x-y|> R}\left| F(u_{n}(x))\chi _{|u_{n}|\le K_{\varepsilon }} \right| ^{p_{3}'}\textrm{d}y \right] ^{\frac{1}{p_{3}'}}\left[ \int _{|x-y|> R}\frac{1}{|y|^{\alpha p_{3}}|x-y|^{\mu p_{3}}}\textrm{d}y \right] ^{\frac{1}{p_{3}}}\\&\quad \le \ \left( \pi R^{2}\right) ^{\frac{1}{p_{2}'}}{\mathcal {O}}\left( R^{2/p_{2}-\alpha -\mu }\right) \max _{|t|\le K_{\varepsilon }}\left| F(t)\right| + {\mathcal {O}}\left( R^{2/p_{3}-\alpha -\mu }\right) \Vert u_{n}\Vert _{\frac{(4-2\alpha -\mu )pp_{3}'}{4}}^{\frac{(4-2\alpha -\mu )p}{4}}\\&\quad \le \ C. \end{aligned}$$

It follows that

$$\begin{aligned} \left| \frac{\zeta _{n}(x)f(u_{n}(x))\varphi (x)\chi _{|u_{n}|\le M_{\varepsilon }}}{|x|^{\alpha }}\right| \le C\left| \frac{\varphi (x)\max _{|t|\le M_{\varepsilon }}\left| f(t)\right| }{|x|^{\alpha }} \right| \le \frac{C'}{|x|^{\alpha }},\ \ \forall \ x\in \Omega . \end{aligned}$$

By $\alpha <2$, it is easy to verify that $\frac{1}{|x|^{\alpha }}\in L_{\textrm{loc}}^{1}({{\mathbb {R}}} ^{2})$. Therefore, together with $\zeta _{n}(x)\rightarrow {\bar{\zeta }}(x)$ and the Lebesgue dominated convergence theorem, we have

$$\begin{aligned}&\int _{(|u_{n}|\le M_{\varepsilon })\cap (|{\bar{u}}|\ne M_{\varepsilon })} \left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(x))\chi _{|u_{n}|\le K_{\varepsilon }}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{|f(u_{n}(x))\varphi (x)|}{|x|^{\alpha }}\textrm{d}x\\&\quad \rightarrow \int _{|{\bar{u}}|<M_{\varepsilon }}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u(x))\chi _{|u|\le K_{\varepsilon }}}{|y|^{\alpha }|x-y|^{\mu }} \textrm{d}y\right) \frac{|f(u(x))\varphi (x)|}{|x|^{\alpha }}\textrm{d}x. \end{aligned}$$

From the arguments above all and by the arbitrariness of $\varepsilon >0$, we can conclude this Lemma. $\square$

Lemma 4.2

Assume that $\{u_{n}\}$ is bounded in $H_r^1({{\mathbb {R}}} ^2)$, $u_{n}\rightharpoonup u$ in $H_r^1({{\mathbb {R}}} ^2)$ and

$$\begin{aligned} \int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{f(u_{n}(x))u_{n}(x)}{|x|^{\alpha }}\textrm{d}x\le C. \end{aligned}$$

(4.1)

Then we have

$$\begin{aligned} \int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u_{n}(x))}{|x|^{\alpha }}\textrm{d}x \rightarrow \int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u(x))}{|x|^{\alpha }}\textrm{d}x. \end{aligned}$$

Proof

In view of $u_n\rightharpoonup {\bar{u}}\ \ \text{ in }\ H_r^1({{\mathbb {R}}} ^2)$, we know $u_{n}\rightarrow u$ in $L^{q}({{\mathbb {R}}} ^{2})$ with $q> 2$. By [35, Theorem A.1], there exists $g\in L^{q}({{\mathbb {R}}} ^{2})$ such that

$$\begin{aligned} |u_{n}(x)|\le g(x), \ \ |u(x)|\le g(x), \ \ \mathrm{a.e.} \ \ x\in {{\mathbb {R}}} ^{2}. \end{aligned}$$

For any given $\varepsilon \in (0,M_{0}/t_{0})$, it follows from (F5) that

$$\begin{aligned}&\int _{|u_{n}|\ge M_{0}\varepsilon ^{-1}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u_{n}(x))}{|x|^{\alpha }}\textrm{d}x\\&\quad \le \ M_{0}\int _{|u_{n}|\ge M_{0}\varepsilon ^{-1}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{|f(u_{n}(x))|}{|x|^{\alpha }}\textrm{d}x\\&\quad \le \ \varepsilon \int _{|u_{n}|\ge M_{0}\varepsilon ^{-1}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{f(u_{n}(x))u_{n}(x)}{|x|^{\alpha }}\textrm{d}x\le C\varepsilon . \end{aligned}$$

Similarly, one has

$$\begin{aligned} \int _{|u|\ge M_{0}\varepsilon ^{-1}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u(x))}{|x|^{\alpha }} \textrm{d}x\le C\varepsilon . \end{aligned}$$

Now, we can choose $R_{\varepsilon }>0$ such that

$$\begin{aligned} \int _{{{\mathbb {R}}} ^{2}\setminus B_{R_{\varepsilon }}}\left| \left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u(x))}{|x|^{\alpha }}\right| \textrm{d}x<\varepsilon , \end{aligned}$$

$$\begin{aligned} \int _{{{\mathbb {R}}} ^{2}\setminus B_{R_{\varepsilon }}}\left| \left( \int _{{{\mathbb {R}}} ^{2}}\frac{|u(y)|^{q}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{|u(x)|^{q}}{|x|^{\alpha }}\right| \textrm{d}x<\varepsilon , \end{aligned}$$

and

$$\begin{aligned} \int _{{{\mathbb {R}}} ^{2}\setminus B_{R_{\varepsilon }}}\left| \left( \int _{{{\mathbb {R}}} ^{2}}\frac{g^{{\tilde{q}}+1}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{g^{{\tilde{q}}+1}}{|x|^{\alpha }}\right| \textrm{d}x<\varepsilon . \end{aligned}$$

Let C be the constant in (4.1) and choose $K \ge \max \{CM_{0}/\varepsilon ,t_{0}\}$ such that

$$\begin{aligned} \int _{|u|\le M_{0}\varepsilon ^{-1}}\left( \int _{|u|\ge K}\frac{F(u(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u(x))}{|x|^{\alpha }} \textrm{d}x<\varepsilon . \end{aligned}$$

(4.2)

By (F5), one has

$$\begin{aligned}&\int _{|u_{n}|\le M_{0}\varepsilon ^{-1}}\left( \int _{|u_{n}|\ge K}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u_{n}(x))\chi _{B_{R_{\varepsilon }}}}{|x|^{\alpha }}\textrm{d}x\nonumber \\&\quad \le \frac{1}{K} \int _{|u_{n}|\le M_{0}\varepsilon ^{-1}}\left( \int _{|u_{n}|\ge K}\frac{F(u_{n}(y))u_{n}(y)}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u_{n}(x))\chi _{B_{R_{\varepsilon }}}}{|x|^{\alpha }}\textrm{d}x\nonumber \\&\quad \le \frac{M_{0}}{K}\int _{|u_{n}|\le M_{0}\varepsilon ^{-1}}\left( \int _{|u_{n}|\ge K}\frac{f(u_{n}(y))u_{n}(y)}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u_{n}(x))\chi _{B_{R_{\varepsilon }}}}{|x|^{\alpha }}\textrm{d}x\nonumber \\&\quad \le \frac{M_{0}}{K}\int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{f(u_{n}(x))u_{n}(x)}{|x|^{\alpha }}\textrm{d}x\le \varepsilon . \end{aligned}$$

(4.3)

By (F2), we know that there exist $C>0$ and ${\tilde{q}}>\frac{(4-2\alpha -\mu ) }{2}$ such that for $|t|\le K$,

$$\begin{aligned} |F( t)|\le C|t|^{{\tilde{q}}+1}. \end{aligned}$$

(4.4)

Thus we have

$$\begin{aligned}&\int _{\{{{\mathbb {R}}} ^{2}\setminus B_{R_{\varepsilon }}\}\cap \{|u_{n}|\le M_{0}\varepsilon ^{-1}\}}\left( \int _{|u_{n}|\le K} \frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u_{n}(x))}{|x|^{\alpha }}\textrm{d}x\\&\quad \le C\int _{{{\mathbb {R}}} ^{2}\setminus B_{R_{\varepsilon }}}\left( \int _{|u_{n}|\le K} \frac{u_{n}^{{\tilde{q}}+1}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{u_{n}^{{\tilde{q}}+1}}{|x|^{\alpha }}\textrm{d}x\\&\quad \le C\int _{{{\mathbb {R}}} ^{2}\setminus B_{R_{\varepsilon }}}\left( \int _{|u_{n}|\le K} \frac{g^{{\tilde{q}}+1}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{g^{{\tilde{q}}+1}}{|x|^{\alpha }}\textrm{d}x\le C\varepsilon , \end{aligned}$$

which leads to

$$\begin{aligned}&\ \left| \int _{\{{{\mathbb {R}}} ^{2}\setminus B_{R_{\varepsilon }}\}\cap \{|u_{n}|\le M_{0}\varepsilon ^{-1}\}}\left[ \left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u_{n}(x))}{|x|^{\alpha }}\right. \right. \\&\left. \left. \quad \quad - \left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u(x))}{|x|^{\alpha }}\right] \textrm{d}x \right| \\&\quad \le \ \left| \int _{\{{{\mathbb {R}}} ^{2}\setminus B_{R_{\varepsilon }}\}\cap \{|u_{n}|\le M_{0}\varepsilon ^{-1}\}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u_{n}(x))}{|x|^{\alpha }}\textrm{d}x \right| \\&\qquad + \left| \int _{\{{{\mathbb {R}}} ^{2}\setminus B_{R_{\varepsilon }}\}\cap \{|u_{n}|\le M_{0}\varepsilon ^{-1}\}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u(x))}{|x|^{\alpha }}\textrm{d}x \right| \\&\quad< \ \varepsilon + \int _{\{{{\mathbb {R}}} ^{2}\setminus B_{R_{\varepsilon }}\}\cap \{|u_{n}|\le M_{0}\varepsilon ^{-1}\}}\left( \int _{|u_{n}|\le K}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u_{n}(x)}{|x|^{\alpha }}\textrm{d}x\\&\qquad + \int _{\{{{\mathbb {R}}} ^{2}\setminus B_{R_{\varepsilon }}\}\cap \{|u_{n}|\le M_{0}\varepsilon ^{-1}\}}\left( \int _{|u_{n}|\ge K}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u_{n}(x))}{|x|^{\alpha }}\textrm{d}x< (2+C)\varepsilon . \end{aligned}$$

On the other hand,

$$\begin{aligned}&\ \left| \int _{B_{R_{\varepsilon }}}\left[ \left( \int _{{{\mathbb {R}}} ^{2}} \frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u_{n}(x))}{|x|^{\alpha }}-\left( \int _{{{\mathbb {R}}} ^{2}} \frac{F(u(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u(x))}{|x|^{\alpha }}\right] \textrm{d}x \right| \\&\quad \le \ 2C\varepsilon + \left| \int _{B_{R_{\varepsilon }}\cap \{|u_{n}|\le M_{0}\varepsilon ^{-1}\}}\left( \frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\alpha }}\textrm{d}y \right) \frac{F(u_{n}(x))}{|x|^{\alpha }}\textrm{d}x\right. \\&\quad \quad \left. -\int _{B_{R_{\varepsilon }}\cap \{|u|\le M_{0}\varepsilon ^{-1}\}}\left( \frac{F(u(y))}{|y|^{\alpha }|x-y|^{\alpha }}\textrm{d}y \right) \frac{F(u(x))}{|x|^{\alpha }}\textrm{d}x \right| . \end{aligned}$$

It remains to prove that as $n\rightarrow \infty$,

$$\begin{aligned}&\int _{\{|u_{n}|\le M_{0}\varepsilon ^{-1}\}}\left( \int _{{{\mathbb {R}}} ^{2}} \frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u_{n}(x))\chi _{B_{R_{\varepsilon }}}}{|x|^{\alpha }}\textrm{d}x\nonumber \\&\quad \rightarrow \int _{\{|u|\le M_{0}\varepsilon ^{-1}\}} \left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u(x))\chi _{B_{R_{\varepsilon }}}}{|x|^{\alpha }}\textrm{d}x. \end{aligned}$$

(4.5)

Combining (4.2) with (4.3), we can see that

$$\begin{aligned}&\left| \int _{|u_{n}|\le M_{0}\varepsilon ^{-1}}\left\{ \int _{|u_{n}|\ge K}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\frac{F(u_{n}(x))\chi _{B_{R_{\varepsilon }}}}{|x|^{\alpha }}\right. \right. \\&\quad \quad \quad \quad \left. \left. -\int _{|u|\ge K}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u(y))}{|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u(x))\chi _{B_{R_{\varepsilon }}}}{|x|^{\alpha }} \right\} \textrm{d}x\right| \le 2\varepsilon . \end{aligned}$$

In order to prove (4.5), it remains to verify that as $n\rightarrow +\infty$ there holds

$$\begin{aligned}&\int _{\{|u_{n}|\le M_{0}\varepsilon ^{-1}\}}\left( \int _{|u_{n}|\le K} \frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u_{n}(x))\chi _{B_{R_{\varepsilon }}}}{|x|^{\alpha }}\textrm{d}x\\&\quad \rightarrow \int _{\{|u|\le M_{0}\varepsilon ^{-1}\}} \left( \int _{|u|\le K}\frac{F(u(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{F(u(x))\chi _{B_{R_{\varepsilon }}}}{|x|^{\alpha }}\textrm{d}x. \end{aligned}$$

Indeed, it can be easily verified that as $n\rightarrow \infty$,

$$\begin{aligned}&\left( \int _{|u_{n}|\le K}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u_{n}(x))}{|x|^{\alpha }}\chi _{\{B_{R_{\varepsilon }}\cap |u_{n}|\le M_{0}\varepsilon ^{-1}\}}\\ \rightarrow&\left( \int _{|u|\le K}\frac{F(u(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u(x))}{|x|^{\alpha }}\chi _{\{B_{R_{\varepsilon }}\cap |u|\le M_{0}\varepsilon ^{-1}\}} \ \ \ \textrm{pointwise } \ \ \mathrm{a.e.} \end{aligned}$$

From (4.4), we have

$$\begin{aligned}&\int _{B_{R_{\varepsilon }}\cap |u_{n}|\le M_{0}\varepsilon ^{-1}}\left( \int _{|u_{n}|\le K}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u_{n}(x))}{|x|^{\alpha }}\textrm{d}x\\&\quad \le C\int _{B_{R_{\varepsilon }}\cap |u_{n}|\le M_{0}\varepsilon ^{-1}}\left( \int _{|u_{n}|\le K}\frac{|u_{n}(y)|^{{\tilde{q}}+1}}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{|u_{n}(x)|^{{\tilde{q}}+1}}{|x|^{\alpha }} \textrm{d}x\\&\quad \le C\int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{|u_{n}(y)|^{{\tilde{q}}+1}(y)}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y \right) \frac{|u_{n}(x)|^{{\tilde{q}}+1}}{|x|^{\alpha }}\textrm{d}x\\&\quad \le C\cdot C(\mu ,\alpha )\Vert u_{n}\Vert _{\frac{4({\tilde{q}}+1)}{4-2\alpha -\mu }}^{2({\tilde{q}}+1)}\rightarrow C\cdot C(\mu ,\alpha )\Vert u\Vert _{\frac{4({\tilde{q}}+1)}{4-2\alpha -\mu }}^{2({\tilde{q}}+1)}, \ \ \textrm{as}\ \ n\rightarrow \infty . \end{aligned}$$

From [10, Theorem 4.9], there exists ${\mathcal {F}}\in L^{1}({{\mathbb {R}}} ^{2})$ such that up to a subsequence, still denoted by $\{u_{n}\}$, for each $n\in {\mathbb {N}}$, we have

$$\begin{aligned} \left| \left( \int _{|u_{n}|\le K}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u_{n}(x))}{|x|^{\alpha }}\chi _{\{B_{R_{\varepsilon }}\cap |u_{n}|\le M_{0}\varepsilon ^{-1}\}}\right| \le |{\mathcal {F}}(x)|. \end{aligned}$$

Using the Lebesgue dominated convergence theorem, we can conclude this Lemma. $\square$

Lemma 4.3

Assume that (F1)–(F3) hold. If there exist $u\in H^1({{\mathbb {R}}} ^2)$ and $\lambda \in {{\mathbb {R}}}$ such that

$$\begin{aligned} -\Delta u+\lambda u=\left( I_{\mu }\times \frac{F(u)}{|x|^{\alpha }}\right) \frac{f(u)}{|x|^{\alpha }}, \ \ \ \ x\in {{\mathbb {R}}} ^2, \end{aligned}$$

then $J(u)=0$, where J is defined by (2.2).

The proof of lemma is standard, so we omit it. Hereafter, we are ready to prove the Theorem 1.4.

Proof of Theorem 1.4:

Let ${\mathcal {S}}_a^r={\mathcal {S}}_a\cap H_r^1({{\mathbb {R}}} ^2)$. In the same way as Lemmas 2.8 and 3.2, we can deduce that for any $a>0$, there exists a bounded sequence $\{u_n\}\subset {\mathcal {S}}_a^r$ such that

$$\begin{aligned} \Phi (u_n)\rightarrow {c}_r(a)\in (0, 2\pi /\alpha _0), \ \ \Phi |_{{\mathcal {S}}_a^r}'(u_n) \rightarrow 0\ \ \text{ and }\ \ J(u_n)\rightarrow 0, \end{aligned}$$

(4.6)

and

$$\begin{aligned} {c}_r(a)&=\inf _{g\in \Gamma _{r,a}}\max _{t\in [0,1]}\Phi (g(t)) >\max _{g\in \Gamma _{r,a}}\max \{\Phi (g(0)),\Phi (g(1))\}, \end{aligned}$$

where $\Gamma _{r,a}=\{g\in {\mathcal {C}}([0,1],{\mathcal {S}}_a^r):\Vert \nabla g(0)\Vert _2^2\le K(a), \Phi (g(1))<0\}$ and K(a) is given in Lemma 2.3. Then there exists ${\bar{u}}\in H_r^1({{\mathbb {R}}} ^2)$ such that, passing to a subsequence,

$$\begin{aligned} u_n\rightharpoonup {\bar{u}}\ \ \text{ in }\ H_r^1({{\mathbb {R}}} ^2), \ \ u_n\rightarrow {\bar{u}}\ \ \text{ in }\ L^s({{\mathbb {R}}} ^2) \ \text{ for } \ s>2, \ \ u_n\rightarrow {\bar{u}}\ \text{ a.e. } \text{ in }\ {{\mathbb {R}}} ^2. \end{aligned}$$

Arguing similar as Lemma 2.8, we know that

$$\begin{aligned} \int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{f(u_{n}(x))u_{n}(x)}{|x|^{\alpha }}\textrm{d}x\le {\mathcal {K}}. \end{aligned}$$

Then, it follows that Lemma 4.2 holds, that is

$$\begin{aligned}{} & {} \int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u_{n}(x))}{|x|^{\alpha }}\textrm{d}x\nonumber \\{} & {} \quad =\int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}} \frac{F({\bar{u}}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F({\bar{u}}(x))}{|x|^{\alpha }}\textrm{d}x+o(1). \end{aligned}$$

(4.7)

Next, we claim that ${\bar{u}}\ne 0$. Otherwise, we suppose that $u_n\rightharpoonup 0$ in $H_r^1({{\mathbb {R}}} ^2)$. Then one has

$$\begin{aligned} \Vert \nabla u_n\Vert ^2&= 2\Phi (u_n)+\int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{F(u_{n}(x))}{|x|^{\alpha }}\textrm{d}x\nonumber \\&= 2{c}_r(a)+o(1):=\frac{(4-2\mu -\alpha )\pi }{\alpha _0}(1-3{\tilde{\varepsilon }})+o(1) \ \ \mathrm{for \ some\ constant } \ {\bar{\varepsilon }}>0. \end{aligned}$$

(4.8)

Choosing $q\in (1,2)$ be such that

$$\begin{aligned} \frac{(1+{\bar{\varepsilon }})(1-3{\bar{\varepsilon }})q}{1-{\bar{\varepsilon }}}<1, \end{aligned}$$

using (F1), we get

$$\begin{aligned} |f(t)|^q\le C_{1}\left[ e^{\alpha _0(1+{\bar{\varepsilon }})qt^2}-1\right] , \ \ \ \ \forall \ |t|\ge 1, \end{aligned}$$

and using (ii) of Lemma 1.1, we get

$$\begin{aligned} \int _{|u_n|\ge 1}|f(u_n)|^{\frac{4q}{4-2\alpha -\mu }}\mathrm{{d}}x\le \int _{|u_n|\ge 1}\left( e^{\frac{4\alpha _{0}(1+{\tilde{\varepsilon }})q\Vert \nabla u_n\Vert ^2}{4-2\alpha -\mu } \left( \frac{u_n}{\Vert \nabla u_n\Vert }\right) ^2}-1\right) \mathrm{{d}}x\le C. \end{aligned}$$

Thus,

$$\begin{aligned}&\int _{|u_n|\ge 1}\left( \int _{{{\mathbb {R}}} ^2}\frac{F(u_n(y))}{|y|^\alpha |x-y|^\mu }dy\right) \frac{f(u_n(x))u_n(x)}{|x|^\alpha }dx \\&\quad \le C_2\left[ \int _{{{\mathbb {R}}} ^2}\left( \int _{{{\mathbb {R}}} ^2}\frac{F(u_n(y))}{|y|^\alpha |x-y|^\mu }dy\right) \frac{F(u_n(x))}{|x|^\alpha }dx\right] ^{1/2} \\&\qquad \times \left[ \int _{{{\mathbb {R}}} ^2}\left( \int _{{{\mathbb {R}}} ^2}\frac{f(u_n(y))u_n(y)\chi _{|u_n|\ge 1}}{|y|^\alpha |x-y|^\mu }dy\right) \frac{f(u_n(x))(u_n(x))\chi _{|u_n|\ge 1}}{|x|^\alpha }dx \right] ^{1/2} \\&\quad \le C_3\left[ \int _{|u_n|\ge 1}|f(u_n)|^{\frac{4q}{4-2\alpha -\mu }}dx\right] ^{\frac{4-2\alpha -\mu }{4q}}\left[ \int _{|u_n|\ge 1}|u_n|^{\frac{4q}{(q-1)(4-2\alpha -\mu )}}dx \right] ^{\frac{(4-2\alpha -\mu )(q-1)}{4q}} \\&\quad \le C_4\Vert u_n\Vert _{\frac{4q}{(q-1)(4-2\alpha -\mu )}}=o(1). \end{aligned}$$

Similarly, we have

$$\begin{aligned} \int _{|u_n|\le 1}\left( \int _{{{\mathbb {R}}} ^2}\frac{F(u_n(y)}{|y|^\alpha |x-y|^\mu }dy\right) \frac{f(u_n(x))u_n(x)}{|x|^\alpha }dx\le C_5\Vert u_n\Vert _2^{(4-2\alpha -\mu )/2}=o(1).\nonumber \\ \end{aligned}$$

(4.9)

Then it follows from (4.9) and (4.9) that

$$\begin{aligned}{} & {} {c}_r(a)+o(1) = \Phi (u_n)-\frac{1}{2}J(u_n) = \left( \int _{{{\mathbb {R}}} ^2}\frac{F(u_n(y))}{|y|^\alpha |x-y|^\mu }dy\right) \\{} & {} \quad \frac{\left[ f(u_n)u_n-\frac{6-\mu -2\alpha }{2}F(u_n)\right] (x)}{2|x|^\alpha }dx=o(1), \end{aligned}$$

which is a contradiction due to ${c}_r(a)>0$ for any $a>0$. This shows that ${\bar{u}}\ne 0$ as claimed.

By (4.6) and the boundedness of the sequence $\{u_n\}$, one can easily verify that there exist a bounded sequence $\{\lambda _n\}\subset {{\mathbb {R}}}$ and ${\bar{\lambda }}$ such that, up to a subsequence,

$$\begin{aligned} \lambda _n \rightarrow {\bar{\lambda }}\in {{\mathbb {R}}} , \end{aligned}$$

(4.10)

$$\begin{aligned} -\Delta u_n+\lambda _n u_n-\left( I_{\mu }\times \frac{F(u_n)}{|x|^{\alpha }}\right) \frac{f(u_n)}{|x|^{\alpha }}\rightarrow 0 \ \ \ \textrm{in} \ \ \ (H_{r}^1({{\mathbb {R}}} ^2))^*, \end{aligned}$$

(4.11)

and

$$\begin{aligned} -\Delta u_n+{\bar{\lambda }} u_n-\left( I_{\mu }\times \frac{F(u_n)}{|x|^{\alpha }}\right) \frac{f(u_n)}{|x|^{\alpha }}\rightarrow 0 \ \ \ \textrm{in} \ \ \ (H_{r}^1({{\mathbb {R}}} ^2))^*. \end{aligned}$$

Again, in view of the conclusion of Lemma 4.1, we can see that

$$\begin{aligned} -\Delta {\bar{u}}+{\bar{\lambda }} {\bar{u}} -\left( I_{\mu }\times \frac{F({\bar{u}})}{|x|^{\alpha }}\right) \frac{f({\bar{u}})}{|x|^{\alpha }}= 0 \ \ \ \textrm{in} \ \ \ (H_{r}^1({{\mathbb {R}}} ^2))^*. \end{aligned}$$

(4.12)

Hereafter, the only thing we need to verify is that $\Vert {\bar{u}}\Vert _2^2=a$, and next our goal is to prove that $u_n\rightarrow {\bar{u}}$ in $H_r^1({{\mathbb {R}}} ^2)$. Note that (4.11) yields

$$\begin{aligned} \Vert \nabla u_n\Vert _2^2+\lambda _n \Vert u_n\Vert _2^2 -\int _{{{\mathbb {R}}} ^2}\left( \frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\right) \frac{f(u_n)u_n}{|x|^{\alpha }}\textrm{d}x\rightarrow 0, \end{aligned}$$

(4.13)

and

$$\begin{aligned} \int _{{{\mathbb {R}}} ^2}\left( \nabla u_n \cdot \nabla {\bar{u}}+\lambda _n u_n{\bar{u}}\right) \textrm{d}x-\int _{{{\mathbb {R}}} ^2}\left( \frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\right) \frac{f(u_n){\bar{u}}}{|x|^{\alpha }}\textrm{d}x\rightarrow 0. \end{aligned}$$

(4.14)

By (4.13) minus $J(u_n)\rightarrow 0$, and using (4.10) and (4.7), we have

$$\begin{aligned} \begin{aligned} {\bar{\lambda }}a+o(1)= \lambda _n \Vert u_n\Vert _2^2=&\frac{6-\mu -2\alpha }{2}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(u_{n})}{|x|^{\alpha }}\right) \frac{F(u_{n})}{|x|^{\alpha }}\textrm{d}x+o(1)\\ =&\frac{6-\mu -2\alpha }{2}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F({\bar{u}})}{|x|^{\alpha }}\right) \frac{F({\bar{u}})}{|x|^{\alpha }}\textrm{d}x+o(1), \end{aligned} \end{aligned}$$

which, together with $F(t)>0$ for $t\ne 0$, yields ${\bar{\lambda }}> 0$. By (4.13) and (4.14), we have

$$\begin{aligned}{} & {} \int _{{{\mathbb {R}}} ^2}\left[ \nabla u_n \cdot \nabla (u_n-{\bar{u}})+\lambda _n u_n(u_n-{\bar{u}})\right] \textrm{d}x\\ {}{} & {} \quad =\int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^2}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{f(u_n)(u_n-{\bar{u}})}{|x|^{\alpha }}\textrm{d}x. \end{aligned}$$

Next, we claim that $\int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^2}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{f(u_n)(u_n-{\bar{u}})}{|x|^{\alpha }}\textrm{d}x=o(1)$. By (4.12) and the Lemma 4.3, we have $J({\bar{u}})=0$. This, jointly with (F3) implies

$$\begin{aligned}{} & {} \Phi ({\bar{u}})=\Phi ({\bar{u}})-\frac{1}{2}J({\bar{u}})=\frac{1}{2}\int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^2}\frac{F({\bar{u}}(y))}{|y|^{\alpha }|x-y|^{\mu }}\right) \\{} & {} \quad \frac{\left[ f({\bar{u}}){\bar{u}}-\frac{6-\mu -2\alpha }{2}F({\bar{u}})\right] (x)}{|x|^{\alpha }}\textrm{d}x\ge 0. \end{aligned}$$

Thus,

$$\begin{aligned} {c}_r(a)+o(1)&= \Phi (u_n) = \frac{1}{2}\Vert \nabla u_n\Vert _2^2-\frac{1}{2}\int _{{{\mathbb {R}}} ^{2}}\left( I_{\mu }\times \frac{F(u_n)}{|x|^{\alpha }}\right) \frac{F(u_n)}{|x|^{\alpha }}\textrm{d}x \\&= \frac{1}{2}\left( \Vert \nabla (u_n- {\bar{u}})\Vert _2^2+\Vert \nabla {\bar{u}}\Vert _2^2\right) \\&-\frac{1}{2}\int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^{2}}\frac{F({\bar{u}}(y))}{|y|^{\alpha }|x-y|^{\mu }} \textrm{d}y\right) \frac{F({\bar{u}}(x))}{|x|^{\alpha }} \textrm{d}x+o(1) \\&= \frac{1}{2}\Vert \nabla (u_n- {\bar{u}})\Vert _2^2+\Phi ({\bar{u}})+o(1) \\&\ge \frac{1}{2}\Vert \nabla (u_n- {\bar{u}})\Vert _2^2+o(1). \end{aligned}$$

Since $0<{c}_r(a)<\frac{(4-2\mu -\beta )\pi }{2\alpha _0}$ for any $a>0$, similarly as in (4.8), it follows that there exists ${\bar{\varepsilon }}>0$ such that

$$\begin{aligned} \Vert \nabla (u_n-{\bar{u}})\Vert _2^2\le \frac{(1-3{\bar{\varepsilon }})^{2}(4-2\mu -\beta )\pi }{\alpha _0} \ \ \text{ for } \text{ large } n\in {\mathbb {N}}. \end{aligned}$$

Nothing that $q/(q-1)>2$, by using the Hölder inequality, we have

$$\begin{aligned} \int _{|u_n|\ge 1}|f(u_n)|^\frac{4q}{4-2\alpha -\mu }\textrm{d}x&\le C_{6}\int _{|u_n|\ge 1}\left[ e^{\frac{\alpha _0(1+{\bar{\varepsilon }})4q u_n^2}{4-2\alpha -\mu }}-1\right] \textrm{d}x\nonumber \\&\le C_6\int _{|u_n|\ge 1}\left[ e^{\frac{4\alpha _0(1+{\bar{\varepsilon }})^2{\bar{\varepsilon }}^{-1}q {\bar{u}}^2}{4-2\alpha -\mu }} e^{\frac{4\alpha _0(1+{\bar{\varepsilon }})^2 q (u_n-{\bar{u}})^2}{4-2\alpha -\mu }}-1\right] \textrm{d}x\nonumber \\&\le \frac{(q-1)C_6}{q}\int _{|u_n|\ge 1}\left[ e^{\frac{\alpha _0(1+{\bar{\varepsilon }})^2{\bar{\varepsilon }}^{-1} q^2(q-1)^{-1} {\bar{u}}^2}{4-2\alpha -\mu }}-1\right] \textrm{d}x\nonumber \\&\ \ \ \ + \frac{C_6}{q}\int _{|u_n|\ge 1}\left[ e^{\frac{4\alpha _0(1+{\bar{\varepsilon }})^2q^2 (u_n-{\bar{u}})^2}{4-2\alpha -\mu }}-1\right] \textrm{d}x\nonumber \\&\le \frac{(q-1)C_6}{q}\int _{{{\mathbb {R}}} ^2}\left[ e^{\frac{4\alpha _0(1+{\bar{\varepsilon }})^2{\bar{\varepsilon }}^{-1} q^2(q-1)^{-1} {\bar{u}}^2}{4-2\alpha -\mu }}-1\right] \textrm{d}x\nonumber \\&\ \ \ \ + \frac{C_6}{q}\int _{{{\mathbb {R}}} ^2}\left[ e^{\frac{4\alpha _0(1+{\bar{\varepsilon }})^2q^2 (u_n-{\bar{u}})^2}{4-2\alpha -\mu }}-1\right] \textrm{d}x \nonumber \\&\le C_7+\frac{C_6}{q}\int _{{{\mathbb {R}}} ^2}\left[ e^{\frac{4\alpha _0(1+{\bar{\varepsilon }})^2q^2 \Vert \nabla (u_n- {\bar{u}})\Vert _2^2 }{4-2\alpha -\mu }\cdot \frac{(u_n-{\bar{u}})^2}{ \Vert \nabla (u_n- {\bar{u}})\Vert _2^2}}-1\right] \textrm{d}x \le C_5, \end{aligned}$$

where $\frac{4\alpha _0(1+{\bar{\varepsilon }})^2q^2 \Vert \nabla (u_n- {\bar{u}})\Vert _2^2 }{4-2\alpha -\mu }< 4\pi (1-3{\bar{\varepsilon }})^{2}(1+{\bar{\varepsilon }})^2q^{2}<4\pi$. Moreover, we have

$$\begin{aligned}&\int _{|u_n|\ge 1}\left( \int _{{{\mathbb {R}}} ^2}\frac{F(u_n(y))}{|y|^\alpha |x-y|^\mu }dy\right) \frac{f(u_n(x))(u_n-{\bar{u}})(x)}{|x|^\alpha }dx \\&\quad \le C_8\left[ \int _{{{\mathbb {R}}} ^2}\left( \int _{{{\mathbb {R}}} ^2}\frac{F(u_n(y))}{|y|^\alpha |x-y|^\mu }dy\right) \frac{F(u_n(x))}{|x|^\alpha }dx\right] ^{1/2} \\&\qquad \times \left[ \int _{|u_n|\ge 1}\left( \int _{|u_n|\ge 1}\frac{f(u_n(y))(u_n-{\bar{u}})(y)}{|y|^\alpha |x-y|^\mu }dy\right) \frac{f(u_n(x))(u_n-{\bar{u}})(x)}{|x|^\alpha }dx \right] ^{1/2} \\&\quad \le C_9\left[ \int _{|u_n|\ge 1}|f(u_n)|^{\frac{4q}{4-2\alpha -\mu }}dx\right] ^{\frac{4-2\alpha -\mu }{4q}}\left[ \int _{|u_n|\ge 1}|u_n-{\bar{u}} |^{\frac{4q}{(q-1)(4-2\alpha -\mu )}}dx\right] ^{\frac{(4-2\alpha -\mu )(q-1)}{4q}} \\&\quad \le C_{10}\Vert u_n-{\bar{u}}\Vert _{\frac{4q}{(q-1)(4-2\alpha -\mu )}}=o(1) , \end{aligned}$$

and similarly one has

$$\begin{aligned} \int _{|u_n|\le 1}\left( \int _{{{\mathbb {R}}} ^2}\frac{F(u_n(y)}{|y|^\alpha |x-y|^\mu }dy\right) \frac{f(u_n(x))(u_n-{\bar{u}})(x)}{|x|^\alpha }dx=o(1). \end{aligned}$$

Till now, we have finished the Claim. Then, one has

$$\begin{aligned}{} & {} \int _{{{\mathbb {R}}} ^2}\left[ \nabla u_n \cdot \nabla (u_n-{\bar{u}})+\lambda _n u_n(u_n-{\bar{u}})\right] \textrm{d}x\\{} & {} \quad =\int _{{{\mathbb {R}}} ^{2}}\left( \int _{{{\mathbb {R}}} ^2}\frac{F(u_{n}(y))}{|y|^{\alpha }|x-y|^{\mu }}\textrm{d}y\right) \frac{f(u_n)(u_n-{\bar{u}})}{|x|^{\alpha }}\textrm{d}x, \end{aligned}$$

which, together with $u_n\rightharpoonup {\bar{u}}$ in $H_r^1({{\mathbb {R}}} ^2)$ and $\lambda \rightarrow {\bar{\lambda }}>0$, implies that $u_n\rightarrow {\bar{u}}$ in $H_r^1({{\mathbb {R}}} ^2)$. Next, using Palais’ principle of symmetric criticality [35], the above function ${\bar{u}}\in H_r^1({{\mathbb {R}}} ^2){\setminus }\{0\}$ is in fact a radial solution of $({\mathcal {P}}_{a})$ in $H^1({{\mathbb {R}}} ^2)$, and so the proof of Theorem 1.4 is completed.

Remark 4.4

We omit here the proof of Theorem 1.6, since the difference between of Theorems 1.4 and 1.6 has been presented in Sect. 3. To prove Theorem 1.6, we just need to replace Lemma 3.2 by Lemma 3.3.