Nonlinear perturbations of a periodic magnetic Choquard equation with Hardy–Littlewood–Sobolev critical exponent

Abstract

In this paper, we consider the following magnetic nonlinear Choquard equation

$$\begin{aligned} -(\nabla +iA(x))^2u+ V(x)u = \left( \frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\right) |u|^{2_{\alpha }^*-2} u + \lambda f(u)\ \text { in }\ \mathbb {R}^N, \end{aligned}$$

where $2_{\alpha }^{*}=\frac{2N-\alpha }{N-2}$ is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, $\lambda >0$, $N\ge 3$, $0<\alpha < N$, $A: \mathbb {R}^{N}\rightarrow \mathbb {R}^{N}$ is an $C^1$, $\mathbb {Z}^N$-periodic vector potential and V is a continuous scalar potential given as a perturbation of a periodic potential. Considering different types of nonlinearities f, namely $f(x,u)=\left( \frac{1}{|x|^{\alpha }}*|u|^{p}\right) |u|^{p-2} u$ for $(2N-\alpha )/N<p<2^{*}_{\alpha }$, then $f(u)=|u|^{p-1} u$ for $1<p<2^*-1$ and $f(u)=|u|^{2^* - 2}u$ (where $2^*=2N/(N-2)$), we prove the existence of at least one ground-state solution for this equation by variational methods if p belongs to some intervals depending on N and $\lambda$.

1 Introduction

In this article, we consider the problem

$$\begin{aligned} -(\nabla +iA(x))^2u+ V(x)u = \left( \frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\right) |u|^{2_{\alpha }^*-2} u + \lambda f(u)\ \text { in }\ {\mathbb {R}}^N, \end{aligned}$$

(1)

where $\nabla +iA(x)$ is the covariant derivative with respect to the $C^1$, ${\mathbb {Z}}^N$-periodic vector potential $A:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}^{N}$, i.e.,

$$\begin{aligned} A(x+y)=A(x),\ \forall \; x\in {\mathbb {R}}^N,\ \forall \; y\in {\mathbb {Z}}^N. \end{aligned}$$

The exponent $2_{\alpha }^{*}=\frac{2N-\alpha }{N-2}$ is critical, in the sense of the Hardy–Littlewood–Sobolev inequality, $\lambda >0$, $N\ge 3,$$0<\alpha < N,$$V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$ is a continuous scalar potential and f stands for different types of nonlinearities. Namely, we first consider $f(x,u)=\left( \frac{1}{|x|^{\alpha }}*|u|^{p}\right) |u|^{p-2} u$ for $(2N-\alpha )/N<p<2^{*}_{\alpha }$ and then $f(u)=|u|^{p-1} u$ for $1<p<2^*-1$, where $2^*$ is the critical exponent of immersion $D^{1,2}({\mathbb {R}}^N)\hookrightarrow L^{2^*}({\mathbb {R}}^N)$, and finally, we examine $f(u)=|u|^{2^* - 2}u$.

Inspired by the seminal work of Coti Zelati and Rabinowitz [17], but also by Alves, Carrião and Miyagaki [1] and by Alves and Figueiredo [2], we assume that there are a continuous, ${\mathbb {Z}}^N$-periodic potential $V_{{\mathcal {P}}}:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$, constants $V_0,W_0>0$ and $W\in L^{\frac{N}{2}}({\mathbb {R}}^N)$ with $W(x)\ge 0$ such that

$(V_1)$:

$V_{{\mathcal {P}}}(x)\ge V_0,\quad \forall \; x\in {\mathbb {R}}^N$,

$(V_2)$:

$V(x)=V_{{\mathcal {P}}}(x)-W(x)\ge W_0,\quad \forall \; x\in {\mathbb {R}}^N$,

where the last inequality is strict on a subset of positive measure in ${\mathbb {R}}^N$.

Since the problem is considered in the whole ${\mathbb {R}}^N$ and has a critical nonlinearity in the Hardy–Littlewood–Sobolev sense, the verification of any compactness condition is not easy.

Our paper is motivated by Gao and Yang in [25], where a classical Choquard equation is considered in a bounded domain, i.e., the case $A\equiv 0$ and $V\equiv 0$ is studied in a bounded domain $\Omega$. There is a huge literature about the Choquard equation, and we cite only Moroz and Van Schaftingen [33] for a good review of results on this important subject. In [25], Gao and Yang proved the existence of a ground-state solution (i.e., a least energy nontrivial solution) under restriction on N and $\lambda$. Other recent advances in the study of the Choquard equation can be found, e.g., in [4, 20, 22, 27, 31, 36] for critical exponents, in [5] for multi-bump solutions and in [6, 7, 32] for the concentration behavior of solutions.

In Mukherjee and Sreenadh [34], the magnetic problem

$$\begin{aligned} -(\nabla +iA(x))^2u+ \mu g(x)u = \ \lambda u+\left( \frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\right) |u|^{2_{\alpha }^*-2} u\ \text { in }\ {\mathbb {R}}^N\end{aligned}$$

was examined. In this equation, $\mu >0$ is also a parameter that interacts with the linear term in the right-hand side of the equation. Existence of a ground-state solution was proved supposing that g satisfies the assumptions

$(g_1)$:

$g\in C({\mathbb {R}}^N, {\mathbb {R}}),$$g\ge 0$ and $\Omega :=$ interior of $g^{-1} (0)$ is a nonempty bounded set with smooth boundary and $g^{-1} (0)={\overline{\Omega }}$;

$(g_2)$:

There exists $M>0$ such that the set $\{x \in {\mathbb {R}}^N\,:\,g(x)\le M\}$ has finite Lebesgue measure in ${\mathbb {R}}^N$.

The concentration of solutions as $\mu \rightarrow \infty$ was also studied.

Changing the right-hand side of (1) to

$$\begin{aligned} \bigg (\frac{1}{|x|^{\alpha }}*|u|^p\bigg )|u|^{p-2}{u}, \end{aligned}$$

(2)

the problem was studied by Cingolani, Clapp and Secchi in [16]. In that paper, the authors proved existence and multiplicity of solutions. In [15], the right-hand side (2) was generalized and a ground-state solution was obtained, but the multiplicity result depends on more restrictive hypotheses than in [16].

Recent years have witnessed a growth of interest in the study of magnetic equations. By using variational methods, penalization techniques and Lyusternik–Schnirelmann theory, in [3], Alves, Figueiredo and Yang proved existence of multiple solutions to the magnetic equation

$$\begin{aligned} \left( \frac{\varepsilon }{i}\nabla -A(x)\right) ^2u+ V(x)u =\varepsilon ^{\mu - N}\left( \frac{1}{|x|^{\alpha }}*F(|u|^2)\right) f(|u|^{2}) u, \quad x\in {\mathbb {R}}^N, \end{aligned}$$

(3)

where $\epsilon >0$ is a parameter, $N\ge 2,$$0< \mu < 2$ and $F(s) =\int \limits _{0}^{s} f (t)\text {d}t$.

The same type of techniques was used by d’Avenia and Ji [18] to obtain multiplicity and concentration of solutions of (3) with the right-hand side of that equation changed to $f(|u|^2)u$, with f having critical exponential growth.

A class of magnetic fractional equations has also verified increasing interest. For example, Fiscella, Pinamonti and Vecchi [24] considered the problem

$$\begin{aligned} (-\Delta )^s_{A}u=\lambda f(|u|)u \quad \text {in }\Omega ,\ \ u=0\quad \text {in }\ {\mathbb {R}}^N\setminus \Omega ,\end{aligned}$$

where $\Omega$ is bounded, $s\in (0,1)$, $\lambda$ is a parameter and

$$\begin{aligned} (-\Delta )^s_{A} u(x)= c_{N,s}\lim _{r\rightarrow 0^+} \int \limits _{B_r^c (x)}\frac{u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)}{|x-y|^{N+2s}}\text {d}y,\end{aligned}$$

where $c_{N,s}$ is a normalizing constant. Considering different types of nonlinearities f, existence of at least two solution was obtained by using variational techniques.

The equation

$$\begin{aligned}(-\Delta )^s_{A}u+u=|u|^{p-2}u\quad \text {in }\ {\mathbb {R}}^3\end{aligned}$$

was considered by d’Avenia and Squassina in [19] where a ground-state solution was obtained by concentration compactness arguments.

In [8], by applying variational methods and Lyusternik–Schnirelmann theory, Ambrosio and d’Avenia proved existence and multiplicity of solutions for the equation

$$\begin{aligned} \varepsilon ^{2s} (-\Delta )^s_{A/ \varepsilon } u + V(x)u=f(|u|^2)u\quad \text {in }\ {\mathbb {R}}^N, \end{aligned}$$

where $\varepsilon >0$ is a parameter, $N\ge 3$, $V \in C({\mathbb {R}}^N, {\mathbb {R}})$ and $A\in C^{0,\alpha }({\mathbb {R}}^N, {\mathbb {R}}^N))$ (for $\alpha \in (0, 1]$) are the electric and magnetic potentials, respectively, and $f:{\mathbb {R}}^N \rightarrow {\mathbb {R}}$ is a subcritical nonlinearity. The same equation with the term $\epsilon ^{-2t} (|x|^{2t-3} * |u|^2)u$ added to the left-hand side of the equation was considered by Ambrosio [12], where $t\in (0,1)$ is a parameter and $N=3$. Existence, multiplicity and concentration of solutions were obtained for $\epsilon >0$ small enough by applying Lyusternik–Schnirelmann theory. See also [11] for a related result.

In [10], Ambrosio investigated existence and concentration of nontrivial solutions to the fractional Choquard equation

$$\begin{aligned} \varepsilon ^{2s} (-\Delta )^2_{A/ \varepsilon } u + V(x)u=\varepsilon ^{\mu - N}\left( \frac{1}{|x|^{\alpha }}*F(|u|^2)\right) f(|u|^{2})u\quad \text {in }\ {\mathbb {R}}^N, \end{aligned}$$

where $\varepsilon >0$ is a parameter, $s \in (0, 1),$$0<\mu <2s,$$N\ge 3,$$(-\Delta ^s_A)$ is the fractional magnetic Laplacian, $A:{\mathbb {R}}^N\rightarrow {\mathbb {R}}^N$ is a smooth magnetic potential, $V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$ is a positive potential with a local minimum and f is a continuous nonlinearity with subcritical growth.

The main results of this paper are the following theorems.

Theorem 1

For $\frac{2N-\alpha }{N}<p<2_{\alpha }^*$, under the hypotheses already stated on A, V and $\alpha$, problem

$$\begin{aligned} -(\nabla +iA(x))^2u+ V(x)u =\left( \frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\right) |u|^{2_{\alpha }^*-2} u + \lambda \left( \frac{1}{|x|^{\alpha }}*|u|^{p}\right) |u|^{p-2} u\ \text { in }\ {\mathbb {R}}^N \end{aligned}$$

(4)

has at least one ground-state solution if either

(i):

$\frac{N+2-\alpha }{N-2}<p<2_{\alpha }^*$, $N=3,4$ and $\lambda >0$;

(ii):

$\frac{2N-\alpha }{N}<p\le \frac{N+2-\alpha }{N-2}$, $N=3,4$ and $\lambda$ sufficiently large;

(iii):

$\frac{2N-2-\alpha }{N-2}<p<2_{\alpha }^*$, $N\ge 5$ and $\lambda >0$;

(iv):

$\frac{2N-\alpha }{N}<p\le \frac{2N-2-\alpha }{N-2}$, $N\ge 5$ and $\lambda$ sufficiently large.

Theorem 2

For $1<p<2^{*} - 1$, under the hypotheses already stated on A, V and $\alpha$, problem

$$\begin{aligned} -(\nabla +iA(x))^2u+ V(x)u = \ \bigg (\frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\bigg )|u|^{2_{\alpha }^*-2} u + \lambda |u|^{p-1}u\ \text { in }\ {\mathbb {R}}^N \end{aligned}$$

has at least one ground-state solution if either

(i):

$3<p<5$, $N=3$ and $\lambda >0;$

(ii):

$p>1$, $N\ge 4$ and $\lambda >0$;

(iii):

$1<p\le 3$, $N=3$ and $\lambda$ sufficiently large.

Theorem 3

Under the hypotheses already stated on A, V and $\alpha$, the problem

$$\begin{aligned} -(\nabla +iA(x))^2u+ V(x)u = \ \lambda \left( \frac{1}{|x|^{\alpha }}*|u|^{p}\right) |u|^{p-2} u+ |u|^{2^* - 2} u\ \text { in }\ {\mathbb {R}}^N \end{aligned}$$

has at least one ground-state solution in the intervals already described in Theorem 1.

Initially, we are going to prove the existence of a ground-state solution for problem (1) considering the potential $V=V_{{\mathcal {P}}}$, that is, we consider the problem

$$\begin{aligned} -(\nabla +iA(x))^2u+ V_{{\mathcal {P}}}(x)u =\left( \frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\right) |u|^{2_{\alpha }^*-2} u + \lambda f(u)\ \text { in }\ {\mathbb {R}}^N \end{aligned}$$

(5)

and f as in Theorems 1, 2 and 3 , where we maintain the notation introduced before and suppose that $(V_1)$ is valid.

As in Gao and Yang in [25], the key step to proof the existence of a ground-state solution of problem (5) is the use of cut-off techniques on the extreme function that attains the best constant $S_{H,L}$ defined in the sequence. This allows us to estimate the mountain pass value $c_{\lambda }$ associated with the energy functional $J_{A,V_{{\mathcal {P}}}}$ related to (5) in terms of the level where the PS condition holds. In a demanding proof, this leads us to establish intervals for p (depending on N and $\lambda$) where the PS condition is satisfied, as in the seminal work of Brézis and Nirenberg [14]. After that, the proof is completed by showing the mountain pass geometry, introducing the Nehari manifold associated with (5) and applying concentration–compactness arguments. In the sequel, we consider (1) for the different nonlinearities f and prove that each problem has at least one ground-state solution.

We observe that the conclusion of Theorem 2 is similar to that of Theorem 1.1 in Alves, Carrião and Miyagaki [1] and Theorem 1.1 in Miyagaki [30]. Being more precise, in [1] the authors have discussed the existence of a positive solution to the semilinear elliptic problem involving critical exponents

$$\begin{aligned}-\Delta u + V(x) u= \lambda u^{q} + u^{p}\ \text { in }\ {\mathbb {R}}^N,\end{aligned}$$

where $\lambda >0$ is a parameter, $1<q<p=2^*-1$ and $V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$ is a positive continuous function. On its turn, Miyagaki [30] has studied the existence of nontrivial solution for the following class of semilinear elliptic equation in ${\mathbb {R}}^N$ ($N\ge 3$) involving critical Sobolev exponents

$$\begin{aligned}-\Delta u + a(x) u= \lambda |u|^{q-1} + |u|^{p-1} u \ \text { in }\ {\mathbb {R}}^N,\end{aligned}$$

where $1<q<p\le 2^*-1=\frac{N+2}{N-2}$ and $\lambda >0$ are constants and $a:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$ is a continuous function such that $a(x)\ge a_0$ for all $x\in {\mathbb {R}}^N$, where $a_0>0$ is a constant.

Problems (5) and (1) are then related by showing that the minimax value $d_\lambda$ of the latter satisfies $d_\lambda <c_\lambda$. Once more, concentration–compactness arguments are applied to show the existence of a ground-state solution.

This paper is organized as follows. In Sect. 2, some preliminary results are established. Sections 3, 4 and 5 are then devoted to the proofs of Theorems  1, 2 and  3, respectively.

2 Preliminary results

We denote

$$\begin{aligned}\nabla _{A} u=\nabla u+iA(x)u.\end{aligned}$$

We handle problem (1) in the space

$$\begin{aligned}H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})=\left\{ u\in L^2({\mathbb {R}}^N,{\mathbb {C}})\,:\, \nabla _{A} u\in L^2({\mathbb {R}}^N,{\mathbb {C}}),\ \ \int \limits _{{\mathbb {R}}^N}V(x) |u(x)|^2\;\text {d}x<\infty \right\} \end{aligned}$$

endowed with the norm

$$\begin{aligned}\Vert u\Vert _{A,V}=\bigg (\int \limits _{{\mathbb {R}}^N}(|\nabla _{A} u|^2+V(x)|u|^2)\;\text {d}x \bigg )^{\frac{1}{2}}.\end{aligned}$$

Observe that the norm generated by this scalar product is equivalent to the norm obtained by considering $V\equiv 1$, see [29, Definition 7.20].

If $u\in H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})$, then $|u|\in H^1({\mathbb {R}}^N)$ and the diamagnetic inequality is valid (see [16] or [29, Theorem 7.21])

$$\begin{aligned}|\nabla |u|(x)|\le |\nabla u(x)+iA(x)u(x)|,\ \ \text {a.e. } x\in {\mathbb {R}}^N.\end{aligned}$$

As a consequence of the diamagnetic inequality, we have the continuous immersion

$$\begin{aligned} H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})\hookrightarrow L^s({\mathbb {R}}^N,{\mathbb {C}})\end{aligned}$$

(6)

for any $s\in [2,\frac{2N}{N-2}]$. We denote $2^*=\frac{2N}{N-2}$ and $\Vert \cdot \Vert _s$ the norm in $L^s({\mathbb {R}}^N,{\mathbb {C}})$.

It is well known that $C^\infty _c({\mathbb {R}}^N,{\mathbb {C}})$ is dense in $H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$, see [29, Theorem 7.22].

Following Gao and Yang [26], we denote by $S_{H,L}$

$$\begin{aligned} S_{H,L}:&=\inf _{u\;\in \; D^{1,2}({\mathbb {R}}^N,{\mathbb {R}})\setminus \{0\}}\frac{\displaystyle \int \limits _{{\mathbb {R}}^N}|\nabla u|^2 \text {d}x}{\left( \displaystyle \int \limits _{{\mathbb {R}}^N}\int \limits _{{\mathbb {R}}^N}\frac{|u(x)^{2_{\alpha }^*}|\ |u(y)|^{2_{\alpha }^*}}{|x-y|^{\alpha }}\text {d}x\text {d}y\right) ^{\frac{N-2}{2N-\alpha }}}\\&=\inf _{u\in D^{1,2}_A({\mathbb {R}}^N)\setminus \{0\}}\frac{\displaystyle \int \limits _{{\mathbb {R}}^N}|\nabla _A u|^2\text {d}x}{\left( \displaystyle \int \limits _{{\mathbb {R}}^N}\int \limits _{{\mathbb {R}}^N}\frac{|u(x)^{2_{\alpha }^*}|\ |u(y)|^{2_{\alpha }^*}}{|x-y|^{\alpha }}\text {d}x\text {d}y\right) ^{\frac{N-2}{2N-\alpha }}}=:S_A,\nonumber \end{aligned}$$

(7)

where $D^{1,2}_A({\mathbb {R}}^N)=\{u\in L^{2^*}({\mathbb {R}}^N,{\mathbb {C}})\})\,:\,\nabla _A u\in L^2({\mathbb {R}}^N,{\mathbb {C}})\}$. The equality between $S_{H,L}$ and $S_A$ was proved in Mukherjee and Sreenadh [34]. We remark that $S_A$ is attained if and only if $rot \, A=0$ [34, Theorem 4.1]. See also [13, Theorem 1.1].

We state a result proved in [26].

Proposition 4

(Gao and Yang [26]). The constant $S_{H,L}$ defined in (7) is achieved if and only if

$$\begin{aligned}u=C\left( \frac{b}{b^2+|x-a|^2}\right) ^{\frac{N-2}{2}},\end{aligned}$$

where $C>0$ is a fixed constant and $a\in {\mathbb {R}}^N$ and $b\in (0,\infty )$ are parameters. Furthermore,

$$\begin{aligned}S_{H,L}=\frac{S}{C (N,\alpha )^{\frac{N-2}{2N-\alpha }}},\end{aligned}$$

where S is the best Sobolev constant of the immersion $D^{1,2}({\mathbb {R}}^N)\hookrightarrow L^{2^*}({\mathbb {R}}^N)$ and $C (N,\alpha )$ depends on N and $\alpha$.

If we consider the minimizer for S given by $U(x):=\frac{[N(N-2)]^{\frac{N-2}{4}}}{(|1+|x|^2|)^{\frac{N-2}{2}}}$ (see [37, Theorem 1.42]), then

$$\begin{aligned}{\bar{U}}(x)=S^{\frac{(N-\alpha )(2-\alpha )}{4(N+2-\alpha )}}C(N,\alpha )^{\frac{2-N}{2(N+2-\alpha )}}\frac{[N(N-2)]^{\frac{N-2}{4}}}{(|1+|x|^2|)^\frac{N-2}{2}}\end{aligned}$$

is the unique minimizer for $S_{H,L}$ that satisfies

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

with

$$\begin{aligned}\displaystyle \int \limits _{{\mathbb {R}}^N}|\nabla {\bar{U}}|^2 \text {d}x=\displaystyle \int \limits _{{\mathbb {R}}^N}\int \limits _{{\mathbb {R}}^N}\frac{|{\bar{U}}(x)|^{2_{\alpha }^*} |{\bar{U}}(y)|^{2_{\alpha }^*}}{|x-y|^{\alpha }}\text {d}x\text {d}y=S_{H,L}^{\frac{2N-\alpha }{N+2-\alpha }}.\end{aligned}$$

Proposition 5

(Hardy–Littlewood–Sobolev inequality, see [29]). Suppose that $f\in L^t({\mathbb {R}}^N)$ and $h \in L^r({\mathbb {R}}^N)$ for $t,r>1$ and $0<\alpha <N$ satisfying $\frac{1}{t}+\frac{\alpha }{N}+\frac{1}{r}=2$. Then, there exists a sharp constant $C(t,N,\alpha ,r)$, independent of f and h, such that

$$\begin{aligned} \int \limits _{{\mathbb {R}}^N}\int \limits _{{\mathbb {R}}^N}\frac{f(x) h(y)}{|x-y|^{\alpha }}\text {d}x\text {d}y\le C(t,N,\alpha ,r)\Vert f\Vert _t \Vert h\Vert _r. \end{aligned}$$

(8)

If $t=r=\frac{2N}{2N-\alpha }$, then

$$\begin{aligned}C(t,N,\alpha ,r)=C(N,\alpha )=\pi ^{\frac{\alpha }{2}}\frac{\Gamma (\frac{N}{2}-\frac{\alpha }{2})}{\Gamma (N-\frac{\alpha }{2})}\left\{ \frac{\Gamma (\frac{N}{2})}{\Gamma (N)}\right\} ^{-1+\frac{\alpha }{N}}.\end{aligned}$$

In this case, there is equality in (8) if and only if $h=cf$ for a constant c and

$$\begin{aligned}f(x)=A(\gamma ^2+|x-a|^2)^{-(2N-\alpha )/2}\end{aligned}$$

for some $A\in {\mathbb {C}}$, $0\ne \gamma \in {\mathbb {R}}$ and $a\in {\mathbb {R}}^N$.

Lemma 6

Let $U\subseteqq {\mathbb {R}}^{N}$ be any open set. For $1<p<\infty$, let $(f_n)$ be a bounded n in $L^s(U,{\mathbb {C}})$ such that $f_n(x)\rightarrow f(x)$ a.e. Then, $f_n\rightharpoonup f$ in $L^s(U,{\mathbb {C}})$.

The proof of Lemma 6 only adapts the arguments given for the real case, as in [28, Lemme 4.8, Chapitre 1].

3 The case $f(x,u)=\left( \frac{1}{|x|^{\alpha }}*|u|^{p}\right) |u|^{p-2} u$

3.1 The periodic problem

In this subsection, we deal with problem (5) for f(x, u) as above, that is,

$$\begin{aligned} -(\nabla +iA(x))^2u+ V_{{\mathcal {P}}}(x)u =\left( \frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\right) |u|^{2_{\alpha }^*-2} u + \lambda \left( \frac{1}{|x|^{\alpha }}*|u|^{p}\right) |u|^{p-2} u, \end{aligned}$$

(9)

where $\frac{2N-\alpha }{N}<p<2_{\alpha }^*$.

We consider the space

$$\begin{aligned}H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})=\left\{ u\in L^2({\mathbb {R}}^N,{\mathbb {C}})\,:\, \nabla _{A} u\in L^2({\mathbb {R}}^N,{\mathbb {C}})\right\} \end{aligned}$$

endowed with scalar product

$$\begin{aligned}\langle u,v\rangle _{A,V_{{\mathcal {P}}}}=\mathfrak {Re}\int \limits _{{\mathbb {R}}^N}\left( \nabla _A u\cdot \overline{\nabla _A v}+V_{{\mathcal {P}}}(x)u{{\bar{v}}}\right) \text {d}x\end{aligned}$$

and therefore

$$\begin{aligned}\Vert u\Vert ^2_{A,V_{{\mathcal {P}}}}=\int \limits _{{\mathbb {R}}^N}\left( |\nabla _A u|^2+V_{{\mathcal {P}}}|u|^2\right) \text {d}x.\end{aligned}$$

We observe that the energy functional $J_{A,V_{{\mathcal {P}}}}$ on $H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$ associated with (9) is given by

$$\begin{aligned} J_{A,V_{{\mathcal {P}}}}(u):=\frac{1}{2}\Vert u\Vert ^2_{A,V_{{\mathcal {P}}}}-\frac{1}{2\cdot 2_{\alpha }^*}D(u)-\frac{\lambda }{2p}B(u), \end{aligned}$$

where

$$\begin{aligned}B(u)=\int \limits _{{\mathbb {R}}^N}\bigg (\frac{1}{|x|^{\alpha }}*|u|^p\bigg )|u|^p\text {d}x= \int \limits _{{\mathbb {R}}^N}\int \limits _{{\mathbb {R}}^N}\frac{|u(x)^p| |u(y)|^p}{|x-y|^{\alpha }}\text {d}x\text {d}y\end{aligned}$$

and

$$\begin{aligned}D(u)=\int \limits _{{\mathbb {R}}^N}\bigg (\frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\bigg )|u|^{2_{\alpha }^*}\text {d}x=\int \limits _{{\mathbb {R}}^N}\int \limits _{{\mathbb {R}}^N}\frac{|u(x)^{2_{\alpha }^*}| |u(y)|^{2_{\alpha }^*}}{|x-y|^{\alpha }}\text {d}x\text {d}y.\end{aligned}$$

Remark 3.1

If $s\in \left[ (2N-\alpha )/N,(2N-\alpha )/(N-2)\right]$ and $r=2N/(2N-\alpha )$, then $2\le sr\le 2^*$. So, for $u \in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$, it follows from (6) that $u\in L^{sr}({\mathbb {R}}^N,{\mathbb {C}})$, that is, $|u|^s \in L^r({\mathbb {R}}^N,{\mathbb {C}})$. Since $\frac{2}{r}+\frac{\alpha }{N}=2$, the Hardy–Littlewood–Sobolev inequality yields

$$\begin{aligned} \int \limits _{{\mathbb {R}}^N}\int \limits _{{\mathbb {R}}^N} \frac{|u(x)|^s |u(y)|^s}{|x-y|^{\alpha }} \text {d}x\text {d}y\le C(N,\alpha )\Vert u\Vert _{sr}^{2s}. \end{aligned}$$

Therefore,

$$\begin{aligned} B(u)\le C_1(N,\alpha )\Vert u\Vert ^{2p}_{pr} \end{aligned}$$

(10)

and

$$\begin{aligned} D(u)\le C_2(N,\alpha )\Vert u\Vert ^{2\cdot 2_{\alpha }^*}_{2^*} \end{aligned}$$

(11)

for constants $C_1(N,\alpha )$ and $C_2(N,\alpha )$. Therefore, $J_{A,V_{{\mathcal {P}}}}$ is well defined for $u \in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$.

Here, as also in [6], $\frac{2N-\alpha }{N}$ is called the lower critical exponent and $2_{\alpha }^*=\frac{2N-\alpha }{N-2}$ the upper critical exponent. This leads us to say that (1) is a critical nonlocal elliptic equation.

Observe that

$$\begin{aligned} S_{A}=\inf _{u\;\in \; D^{1,2}_A({\mathbb {R}}^N)\setminus \{0\}}\frac{\int \limits _{{\mathbb {R}}^N}|\nabla _A u|^2 \text {d}x}{D(u)^{\frac{N-2}{2N-\alpha }}}. \end{aligned}$$

(12)

Definition 3.1

A function $u\in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$ is a weak solution of (9) if

$$\begin{aligned}\langle u,\psi \rangle _{A,V_{{\mathcal {P}}}}-\mathfrak {Re}\int \limits _{{\mathbb {R}}^N}\left( \frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\right) |u|^{2_{\alpha }^*-2}u{\bar{\psi }}\,\text {d}x-\lambda \; \mathfrak {Re}\int \limits _{{\mathbb {R}}^N}\left( \frac{1}{|x|^{\alpha }}*|u|^p\right) |u|^{p-2}u{\bar{\psi }}\,\text {d}x=0\end{aligned}$$

for all $\psi \in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$.

Since the derivative of the energy functional $J_{A,V_{{\mathcal {P}}}}$ is given by

$$\begin{aligned} J'_{A,V_{{\mathcal {P}}}}(u)\cdot \psi&=\langle u,\psi \rangle _{A,V_{{\mathcal {P}}}}-{\mathfrak {Re}}\int \limits _{{\mathbb {R}}^N}\bigg (\frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\bigg )|u|^{2_{\alpha }^*-2}u{\bar{\psi }}\,\text {d}x-\lambda \; {\mathfrak {Re}}\int \limits _{{\mathbb {R}}^N}\bigg (\frac{1}{|x|^{\alpha }}*|u|^p\bigg )|u|^{p-2}u{\bar{\psi }}\,\text {d}x, \end{aligned}$$

we see that critical points of $J_{A,V_{{\mathcal {P}}}}$ are weak solutions of (9).

Note that if $\psi =u$, we obtain

$$\begin{aligned} J'_{A,V_{{\mathcal {P}}}}(u)\cdot u:=\Vert u\Vert _{A,V_{{\mathcal {P}}}}^{2}-D(u)-\lambda B(u). \end{aligned}$$

(13)

Lemma 7

The functional $J_{A,V_{{\mathcal {P}}}}$ satisfies the mountain pass geometry. Precisely,

(i):

there exists $\rho ,\delta >0$ such that $J_{A,V_{{\mathcal {P}}}}\big |_S\ge \delta >0$ for any $u\in {\mathcal {S}}$, where

$$\begin{aligned}{\mathcal {S}}=\{u\in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})\,:\, \Vert u\Vert _{A,V_{{\mathcal {P}}}}=\rho \};\end{aligned}$$

(ii):

for any $u_0\in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})\setminus \{0\}$ there exists $\tau \in (0,\infty )$ such that $\Vert \tau u_0\Vert _{V_{{\mathcal {P}}}}>\rho$ and $J_{A,V_{{\mathcal {P}}}}(\tau u_0) <0$.

Proof

Inequalities (10) and (11) yield

$$\begin{aligned}J_{A,V_{{\mathcal {P}}}}(u)\ge \frac{1}{2}\Vert u\Vert _{A,V_{{\mathcal {P}}}}^2-\frac{C_2(\alpha ,N)}{2\cdot 2_{\alpha }^*}\Vert u\Vert _{A,V_{{\mathcal {P}}}}^{2\cdot 2_{\alpha }^*}-\frac{\lambda C_1(\alpha ,N)}{2p}\Vert u\Vert _{A,V_{{\mathcal {P}}}}^{2p},\end{aligned}$$

thus implying (i) if we take $\Vert u\Vert _{A,V_{{\mathcal {P}}}}=\rho >0$ sufficiently small.

In order to prove (ii), fix $u_0\in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})\setminus \{0\}$ and consider the function $g_{u_0}:(0,\infty )\rightarrow {\mathbb {R}}$ given by

$$\begin{aligned}g_{u_0}(t):=J_{A,V_{{\mathcal {P}}}}(tu_0)=\frac{1}{2} \Vert tu_0\Vert ^2_{A,V_{{\mathcal {P}}}}- \frac{1}{2\cdot 2^*_{\alpha }}D(t u_0)-\frac{\lambda }{2p}B(t u_0).\end{aligned}$$

We have

$$\begin{aligned} B(tu_0)=&t^{2p}\int \limits _{{\mathbb {R}}^N}\int \limits _{{\mathbb {R}}^N}\frac{|u_0(x)^p| |u_0(y)|^p}{|x-y|^{\alpha }}\text {d}x\text {d}y=t^{2p}B(u_0) \end{aligned}$$

and

$$\begin{aligned} D(tu_0)=&t^{2\cdot 2^*_{\alpha }}\int \limits _{{\mathbb {R}}^N}\int \limits _{{\mathbb {R}}^N}\frac{| u_0(x)|^{2_{\alpha }^*} |u_0(y)|^{2_{\alpha }^*}}{|x-y|^{\alpha }}\text {d}x\text {d}y=t^{2 \cdot 2_{\alpha }^*}D(u_0). \end{aligned}$$

Thus,

$$\begin{aligned} g_{u_0} (t)&=\frac{1}{2} t^2\Vert u_0\Vert ^2_{A,V_{{\mathcal {P}}}}- \frac{1}{2\cdot 2^*_{\alpha }}t^{2\cdot 2^*_{\alpha }}D( u_0)-\frac{\lambda }{2p}t^{2p}B( u_0)\\&=\frac{1}{2} t^{2\cdot 2^*_{\alpha }}\left( \frac{\Vert u_0\Vert ^2_{A,V_{{\mathcal {P}}}}}{t^{(2( 2^*_{\alpha }-1))}}- \frac{1}{2^*_{\alpha }} D( u_0)-\frac{\lambda }{p}\frac{B( u_0)}{t^{(2(2^*_{\alpha }-p)) }}\right) . \end{aligned}$$

Since $1<\frac{2N-\alpha }{N}<p<2^*_{\alpha }$, we have

$$\begin{aligned}\lim _{t\rightarrow +\infty }J^s_{A,V_{{\mathcal {P}}}}(t u_0)=-\infty \end{aligned}$$

completing the proof of (ii). $\square$

The mountain pass theorem without the PS condition (see [37, Theorem 1.15]) yields a Palais–Smale sequence $(u_n)\subset H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})$ such that

$$\begin{aligned}J'_{A,V_{{\mathcal {P}}}}(u_n)\rightarrow 0\qquad \text {and}\qquad J_{A,V_{{\mathcal {P}}}}(u_n)\rightarrow c_{\lambda },\end{aligned}$$

where

$$\begin{aligned} c_{\lambda }=\inf _{\alpha \in \Gamma }\max _{t\in [0,1]}J_{A,V_{{\mathcal {P}}}}(\gamma (t)), \end{aligned}$$

and $\Gamma =\left\{ \gamma \in C^1\left( [0,1],H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})\right) \,:\,\gamma (0)=0,\, J_{A,V_{{\mathcal {P}}}}(\gamma (1))<0\right\}$.

Lemma 8

Suppose that $u_n\rightharpoonup u_0$ in $H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$. Then,

$$\begin{aligned} \frac{1}{|x|^{\alpha }}*|u_n|^{s}\rightharpoonup \frac{1}{|x|^{\alpha }}*|u_0|^{s}\;\; \text {in}\;\;L^{\frac{2N}{\alpha }}({\mathbb {R}}^N), \end{aligned}$$

(14)

for all $\frac{2N-\alpha }{N}\le s\le 2_{\alpha }^*$.

Proof

In this proof, we adapt some ideas of [7]. We can suppose that $|u_n(x)|^s \rightarrow |u_0(x)|^s$ a.e. in ${\mathbb {R}}^N$ and, as a consequence of the immersion (6), $|u_n|^s$ is bounded in $L^{\frac{2N}{2N-\alpha }}({\mathbb {R}}^N)$. Thus, Lemma 6 allows us to conclude that

$$\begin{aligned}|u_n(x)|^s\rightharpoonup |u_0(x)|^s\;\;\text {in}\;L^{\frac{2N}{2N-\alpha }}({\mathbb {R}}^N,{\mathbb {C}})\end{aligned}$$

as $n\rightarrow \infty$.

By the Hardy–Littlewood–Sobolev inequality, the map $T:L^{\frac{2N}{2N-\alpha }}({\mathbb {R}}^N,{\mathbb {C}})\rightarrow L^{\frac{2N}{\alpha }}({\mathbb {R}}^N,{\mathbb {C}})$ defined by $T(w)=|x|^{-\alpha }*w$ is well defined; moreover, it is linear and continuous. Hence, the result follows by applying [21, Proposition 2.1.27]. $\square$

Corollary 9

Suppose that $u_n\rightharpoonup u_0$ and consider

$$\begin{aligned}B'(u_n)\cdot \psi =\mathfrak {Re}\int \limits _{{\mathbb {R}}^N}\bigg (\frac{1}{|x|^{\alpha }}*|u_n|^p\bigg )|u_n|^{p-2}u_n{\bar{\psi }}\,\text {d}x\end{aligned}$$

and

$$\begin{aligned}D'(u_n)\cdot \psi =\mathfrak {Re}\int \limits _{{\mathbb {R}}^N}\bigg (\frac{1}{|x|^{\alpha }}*|u_n|^{2_{\alpha }^*}\bigg )|u_n|^{2_{\alpha }^*-2}u_n{\bar{\psi }}\,\text {d}x,\end{aligned}$$

for $\psi \in C^\infty _c({\mathbb {R}}^{N},{\mathbb {C}})$. Then, $B'(u_n)\cdot \psi \rightarrow B'(u_0)\cdot \psi$ and $D'(u_n)\cdot \psi \rightarrow D'(u_0)\cdot \psi$.

Proof

Immersion (6) guarantees that $|u_n|^{p - 2}u_n$ is bounded in $L^{\frac{2N}{N+2-\alpha }}({\mathbb {R}}^N, {\mathbb {C}})$. Since we can suppose that $|u_n(x)|^p \rightarrow |u_0(x)|^p$ a.e. in ${\mathbb {R}}^N$, by applying Lemma 6, we conclude that

$$\begin{aligned} |u_n|^{p- 2}u_n\rightharpoonup |u_0|^{p - 2}u\quad \;\text {in}\;\; L^{\frac{2N}{N+2-\alpha }}({\mathbb {R}}^N, {\mathbb {C}}) \end{aligned}$$

(15)

for all $\frac{2N-\alpha }{N}\le p\le 2_{\alpha }^*$, as $n\rightarrow +\infty$.

Combining (14) with (15) yields

$$\begin{aligned}\left( \frac{1}{|x|^{\alpha }}*|u_n|^{p}\right) |u_n|^{p-2}u_n\rightharpoonup \left( \frac{1}{|x|^{\alpha }}*|u_0|^{p}\right) |u_0|^{p-2}u_0\;\; \text {in}\;\;L^{\frac{2N}{N+2}}({\mathbb {R}}^N)\end{aligned}$$

as $n\rightarrow +\infty$, for all $\frac{2N-\alpha }{N}\le p\le 2_{\alpha }^*$. Consequently, for $\psi \in C^\infty _c({\mathbb {R}}^{N},{\mathbb {C}})$, it follows that

$$\begin{aligned}\mathfrak {Re}\int \limits _{{\mathbb {R}}^N}\bigg (\frac{1}{|x|^{\alpha }}*|u_n|^p\bigg )|u_n|^{p-2}u_n{\bar{\psi }}\; \text {d}x \rightarrow \mathfrak {Re} \int \limits _{{\mathbb {R}}^N}\bigg (\frac{1}{|x|^{\alpha }}*|u_0|^p\bigg )|u_0|^{p-2}u_0{\bar{\psi }}\; \text {d}x\end{aligned}$$

and

$$\begin{aligned}\mathfrak {Re}\int \limits _{{\mathbb {R}}^N}\bigg (\frac{1}{|x|^{\alpha }}*|u_n|^{2_{\alpha }^*}\bigg )|u_n|^{2_{\alpha }^*-2}u_n{\bar{\psi }}\; \text {d}x \rightarrow \mathfrak {Re} \int \limits _{{\mathbb {R}}^N}\bigg (\frac{1}{|x|^{\alpha }}*|u_0|^{2_{\alpha }^*}\bigg )|u_0|^{2_{\alpha }^*-2}u_0{\bar{\psi }}\; \text {d}x,\end{aligned}$$

that is,

$$\begin{aligned}B'(u_n)\cdot \psi \rightarrow B'(u_0)\cdot \psi \qquad \text {and}\qquad D'(u_n)\cdot \psi \rightarrow D'(u_0)\cdot \psi .\end{aligned}$$

$\square$

Lemma 10

If $(u_n)\subset H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})$ is a Palais–Smale sequence for $J_{A,V_{{\mathcal {P}}}}$, then $(u_n)$ is bounded. In addition, if $u_n\rightharpoonup u$ weakly in $H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})$ as $n\rightarrow \infty$, then u is a weak solution to problem (9).

Proof

Standard arguments prove that $(u_n)$ is bounded in $H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})$. Then, up to a subsequence, we have $u_n\rightharpoonup u$ weakly in $H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})$ as $n\rightarrow \infty$.

From Corollary 9, it follows that for all $\psi \in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})$, we have

$$\begin{aligned}\mathfrak {Re}\int \limits _{{\mathbb {R}}^N}\bigg (\frac{1}{|x|^{\alpha }}*|u_n|^p\bigg )|u|^{p-2}u_n{\bar{\psi }}\; \text {d}x = \mathfrak {Re} \int \limits _{{\mathbb {R}}^N}\bigg (\frac{1}{|x|^{\alpha }}*|u|^p\bigg )|u|^{p-2}u{\bar{\psi }}\; \text {d}x+o_n(1),\ \text { as }\ n\rightarrow \infty , \end{aligned}$$

where $s=p$ or $s=2_{\alpha }^*$.

Thus, since for all $\psi \in C^\infty _c({\mathbb {R}}^{N},{\mathbb {C}})$ we have $J'_{A,V_{{\mathcal {P}}}}(u_n)\cdot \psi = o_n (1)$, we obtain

$$\begin{aligned}J'_{A,V}(u)\cdot \psi =0,\;\;\forall \;\psi \in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}}), \end{aligned}$$

that is, u is a weak solution of (9). $\square$

We now consider the Nehari manifold associated with the $J_{A,V_{{\mathcal {P}}}}$.

$$\begin{aligned}{\mathcal {M}}_{A,V_{{\mathcal {P}}}}&=\left\{ u\in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})\setminus \{0\}\,:\,\Vert u\Vert _{A,V_{{\mathcal {P}}}}^{2}=D(u)+\lambda B(u)\right\} . \end{aligned}$$

Lemma 11

There exists a unique $t_u=t_u (u)>0$ such that $t_u u\in {\mathcal {M}}_{A,V_{{\mathcal {P}}}}$ for all $u\in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})\setminus \{0\}$ and $J_{A,V_{{\mathcal {P}}}}(t_u u)=\displaystyle \max _{t\ge 0} J_{A,V_{{\mathcal {P}}}}(tu)$. Moreover, $c_{\lambda }=c^*_{\lambda }=c^{**}_{\lambda }$, where

$$\begin{aligned}c_{\lambda }^{*}= \displaystyle \inf _{u \;\in \; {\mathcal {M}}_{A,V}} J_{A,V_{{\mathcal {P}}}}(u)\quad \text {and}\quad c_{\lambda }^{**}=\displaystyle \inf _{u\;\in \; H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})\setminus \{0\}}\max _{t\ge 0} J_{A,V_{{\mathcal {P}}}}(tu).\end{aligned}$$

Proof

Let $u\in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})\setminus \{0\}$ and $g_u$ defined on $(0,+\infty )$ given by

$$\begin{aligned} g_u(t)=J_{A,V_{{\mathcal {P}}}}(tu).\end{aligned}$$

By the mountain pass geometry (Lemma 7), there exists $t_u>0$ such that

$$\begin{aligned}g_u(t_u)=\displaystyle \max _{t\ge 0} g_u(t)= \displaystyle \max _{t\ge 0} J_{A,V_{{\mathcal {P}}}}(t_u u).\end{aligned}$$

Hence,

$$\begin{aligned}0=g'_u(t_u)=J'_{A,V_{{\mathcal {P}}}}(t_u u)\cdot u= J'_{A,V_{{\mathcal {P}}}}(t_u u)\cdot t_u u,\end{aligned}$$

implying that $t_u u \in {\mathcal {M}}_{A,V_{{\mathcal {P}}}}$, as a consequence of (13). We now show that $t_u$ is unique. To this end, we suppose that there exists $s_u>0$ such that $s_u u \in {\mathcal {M}}_{A,V_{{\mathcal {P}}}}$. Thus, we have both

$$\begin{aligned} \Vert u\Vert _{A,V_{{\mathcal {P}}}}^2=t_u^{2(2_{\alpha }^*-1)}D(u)+\lambda t_u^{2(p-1)}B(u)\qquad \text {and}\qquad \Vert u\Vert _{A,V_{{\mathcal {P}}}}^2=s_u^{2(2_{\alpha }^*-1)}D(u)+\lambda s_u^{2(p-1)}B(u). \end{aligned}$$

Hence

$$\begin{aligned}0=\left( t_u^{2(2_{\alpha }^*-1)}-s_u^{2(2_{\alpha }^*-1)}\right) D(u)+\lambda \left( t_u^{2(p-1)}-s_u^{2(p-1)}\right) B(u).\end{aligned}$$

Since both terms in parentheses have the same sign if $t_u\ne s_u$ and we also have $B(u)>0$, $D(u)>0$ and $\lambda >0$, it follows that $t_u=s_u$.

Now, the rest of the proof follows arguments similar to that found in [1, 23, 35, 37]. $\square$

Taking into account Lemma 11, we can now redefine a ground-state solution.

Definition 3.2

We say that $u \in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$ is a ground state for problem (9) if $J'_{A,V_{{\mathcal {P}}}}(u)=0$ and $J_{A,V_{{\mathcal {P}}}}(u)=c_{\lambda }$, that is, if u is a solution to the equation $J'_{A,V_{{\mathcal {P}}}}(u)=0$ which has minimal energy in the set of all nontrivial solutions.

The following result controls the level ${c_{\lambda }}$ of a Palais–Smale sequence of $J_{A,V_{{\mathcal {P}}}}$.

Lemma 12

Let $(u_n)\subset H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$ be a $(PS)_{c_{\lambda }}$ sequence for $J_{A,V_{{\mathcal {P}}}}$ such that

$$\begin{aligned} u_n \rightharpoonup 0\quad \text {weakly in}\; H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}}), \;\text { as}\; n\rightarrow \infty , \end{aligned}$$

with

$$\begin{aligned} c_{\lambda }<\frac{N+2-\alpha }{2(2N- \alpha )}S_{A}^{\frac{2N-\alpha }{N-\alpha +2}}. \end{aligned}$$

Then, the sequence $(u_n)$ verifies either

(i):

$u_n\rightarrow 0$ strongly in $H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}}),$ as $n\rightarrow \infty ,$

or

(ii):

There exist a sequence $(y_n)\subset {\mathbb {R}}^N$ and constants $r,\theta >0$ such that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\int \limits _{B_r(y_n)}|u_n|^2 \; \text {d}x\ge \theta , \end{aligned}$$

where $B_r(y)$ denotes the ball in ${\mathbb {R}}^N$ of center at y and radius $r>0$.

Proof

Suppose that (ii) does not hold. Applying a result by Lions [37, Lemma 1.21], it follows from inequality (10) that

$$\begin{aligned} B(u_n)\rightarrow 0,\quad \text {as}\quad n\rightarrow \infty . \end{aligned}$$

Since $J'_{A,V_{{\mathcal {P}}}}(u_n)u_n=o_n(1)$ as $n\rightarrow \infty$, we obtain

$$\begin{aligned} \Vert u_n\Vert ^2_{A,V_{{\mathcal {P}}}}=D(u_n)+o_n (1)\;\;\text {as}\;\; n\rightarrow \infty . \end{aligned}$$

(16)

Let us suppose that

$$\begin{aligned} \Vert u_n\Vert ^2_{A,V_{{\mathcal {P}}}}\rightarrow \ell \ \ (\ell >0)\quad \text {as}\quad n\rightarrow \infty . \end{aligned}$$

Thus, as a consequence of (16), we have

$$\begin{aligned} D(u_n)\rightarrow \ell ,\quad \text {as}\quad n\rightarrow \infty . \end{aligned}$$

Since

$$\begin{aligned}J_{A,V_{{\mathcal {P}}}}(u_n)=\frac{1}{2}\Vert u\Vert ^2_{A,V_{{\mathcal {P}}}}-\frac{\lambda }{2p}B(u_n)-\frac{1}{2\cdot 2_{\alpha }^*}D(u_n),\end{aligned}$$

making $n\rightarrow \infty$ yields

$$\begin{aligned} c_{\lambda }=\frac{\ell }{2}\left( 1-\frac{1}{ 2_{\alpha }^*}\right) =\ell \left( \frac{N+2-\alpha }{ 2(2N-\alpha )}\right) . \end{aligned}$$

(17)

On the other hand, it follows from (12) that

$$\begin{aligned} \Vert u_n\Vert ^2_{A,V_{{\mathcal {P}}}}\ge \displaystyle \int \limits _{{\mathbb {R}}^N} |\nabla _A u_n|^2\; \text {d}x\ge S_{A}(D(u_n))^{\frac{N-2}{2N-\alpha }}, \quad \forall \; u\in D^{1,2}_A({\mathbb {R}}^N). \end{aligned}$$

Thus,

$$\begin{aligned} \ell \ge (S_{A})^{\frac{2N-\alpha }{N+2-\alpha }}, \end{aligned}$$

(18)

and from (17) and (18), we conclude that $c_{\lambda }\ge \frac{N+2-\alpha }{2(2N- \alpha )}S_{A}^{\frac{2N-\alpha }{N+2-\alpha }}$, which is a contradiction. Therefore, (i) is valid and the proof is complete. $\square$

We now state our result about the periodic problem (9).

Theorem 13

Under the hypotheses already stated on A and $\alpha$, suppose that $(V_1)$ is valid. Then, problem (9) has at least one ground-state solution if either

(i):

$\frac{N+2-\alpha }{N-2}<p<2_{\alpha }^*$, $N=3,4$ and $\lambda >0$;

(ii):

$\frac{2N-\alpha }{N}<p\le \frac{N+2-\alpha }{N-2}$, $N=3,4$ and $\lambda$ sufficiently large;

(iii):

$\frac{2N-\alpha -2}{N-2}<p<2_{\alpha }^*$, $N\ge 5$ and $\lambda >0$;

(iv):

$\frac{2N-\alpha }{N}<p\le \frac{2N-\alpha -2}{N-2}$, $N\ge 5$ and $\lambda$ sufficiently large.

Proof

Let $c_\lambda$ be the mountain pass level and consider a sequence $(u_n)\subset H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})$ such that

$$\begin{aligned}J'_{A,V_{{\mathcal {P}}}}(u_n)\rightarrow 0\qquad \text {and}\qquad J_{A,V_{{\mathcal {P}}}}(u_n)\rightarrow c_{\lambda }.\end{aligned}$$

Claim. We affirm that $c_{\lambda }< \frac{N+2-\alpha }{2(2N-\alpha )}(S_{A})^{\frac{2N-\alpha }{N+2-\alpha }}$, a result that will be shown after completing our proof, since it is very technical.

Lemma 10 guarantees that $(u_n)$ is bounded. So, passing to a subsequence if necessary, there is $u\in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})$ such that

$$\begin{aligned} u_n\rightharpoonup u\ \ \text {in}\ \ H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}}), \qquad u_n\rightarrow u\ \ \text {in}\ \ L^2_{loc}({\mathbb {R}}^N,{\mathbb {C}})\qquad \text {and}\qquad u_n\rightarrow u\ \ \text {a.e.}\ \ x\;\in {\mathbb {R}}^N.\end{aligned}$$

(19)

If $u=0$, it follows from Lemma 12 the existence of $\theta >0$ and $(y_n)\subset {\mathbb {R}}^N$ such that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\int \limits _{B_r(y_n)}|u_n|^2 \; \text {d}x\ge \theta . \end{aligned}$$

(20)

A direct computation shows that we can assume that $(y_n)\subset {\mathbb {Z}}^N$. Let

$$\begin{aligned}v_n(x):=u_n (x+y_n).\end{aligned}$$

Since both $V_{{\mathcal {P}}}$ and A are ${\mathbb {Z}}^N$-periodic, we have

$$\begin{aligned}\Vert v_n\Vert _{A,V_{{\mathcal {P}}}}=\Vert u_n\Vert _{A,V_{{\mathcal {P}}}}\quad J_{A,V_{{\mathcal {P}}}}(v_n)= J_{A,V_{{\mathcal {P}}}}(u_n)\quad \text {and}\quad J'_{A,V_{{\mathcal {P}}}}(v_n)\rightarrow 0,\ \ \text {as}\ \ n\rightarrow \infty .\end{aligned}$$

Therefore, there exists $v\in H^1_{A,V_{{\mathcal {P}}}}$ such that $v_n \rightharpoonup v$ weakly in $H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$ and $v_n\rightarrow v$ in $L^2_{loc}({\mathbb {R}}^N,{\mathbb {C}})$.

We claim that $v\ne 0$. In fact, it follows from (20)

$$\begin{aligned} 0<\theta \le \Vert v_n\Vert _{L^2 (B_r(0))}\le \Vert v_n-v\Vert _{L^2 (B_r(0))}+\Vert v\Vert _{L^2 (B_r(0))}. \end{aligned}$$

Since $v_n\rightarrow v$ in $L^2_{loc}({\mathbb {R}}^N)$, we have $\Vert v_n-v\Vert _{L^2 (B_r(0))}\rightarrow 0$ as $n\rightarrow \infty$, proving our claim.

But Corollary 9 guarantees that $J'_{A,V_{{\mathcal {P}}}}(v_n)\cdot \psi \rightarrow J'_{A,V_{{\mathcal {P}}}}(v_n)\cdot \psi$ and it follows that $J'_{A,V_{{\mathcal {P}}}}(v)\cdot \psi =0$. Consequently, v is a weak solution of (9).

Since $v\in {\mathcal {M}}_{A,V_{{\mathcal {P}}}}$, of course we have $c^*_{\lambda }\le J_{A,V_{{\mathcal {P}}}}(v)$. But

$$\begin{aligned} c_{\lambda }^*=c_{\lambda }&= J_{A,V_{{\mathcal {P}}}}(v_n)-\frac{1}{2}J'_{A,V_{{\mathcal {P}}}}(v_n)\cdot u_n + o_n(1)\\&=\lambda \left( \frac{1}{2}-\frac{1}{2p}\right) B(v_n)-\frac{N+2-\alpha }{2(2N-\alpha )}D(v_n)+o_n(1). \end{aligned}$$

Fatou’s lemma then guarantees that as $n\rightarrow \infty$, we have

$$\begin{aligned} c_{\lambda }^*&\ge \lambda \left( \frac{1}{2}-\frac{1}{2p}\right) B(v)-\frac{N+2-\alpha }{2(2N-\alpha )}D(v)= J_{A,V_{{\mathcal {P}}}}(v), \end{aligned}$$

that is, $J_{A,V_{{\mathcal {P}}}}(v)=c_\lambda$, and we are done. The same argument applies to the case $u\ne 0$ in (19). $\square$

We now prove the postponed Claim, that is, we show that $c_{\lambda }< \frac{N+2-\alpha }{2(2N-\alpha )}(S_{A})^{\frac{2N-\alpha }{N+2-\alpha }}$. Observe that once proved the existence of $u_\epsilon$ as in our next result, then

$$\begin{aligned}0<c_{\lambda }=\inf _{\alpha \in \Gamma }\max _{t\in [0,1]}J_{A,V_{{\mathcal {P}}}}(\gamma (t))\le \sup _{t\ge 0}J_{A,V_{{\mathcal {P}}}}(tu_{\varepsilon })<\frac{N+2-\alpha }{2(2N-\alpha )}(S_{A})^{\frac{2N-\alpha }{N+2-\alpha }}. \end{aligned}$$

Lemma 14

There exists $u_{\varepsilon }$ such that

$$\begin{aligned} \sup _{t\ge 0}J_{A,V_{{\mathcal {P}}}}(tu_{\varepsilon })<\frac{N+2-\alpha }{2(2N-\alpha )}(S_{A})^{\frac{2N-\alpha }{N+2-\alpha }}, \end{aligned}$$

provided that either

(i):

$\frac{N+2-\alpha }{N-2}<p<2_{\alpha }^*$, $N=3,4$ and $\lambda >0$;

(ii):

$\frac{2N-\alpha }{N}<p\le \frac{N+2-\alpha }{N-2}$, $N=3,4$ and $\lambda$ sufficiently large;

(iii):

$\frac{2N-2-\alpha }{N-2}<p<2_{\alpha }^*$, $N\ge 5$ and $\lambda >0$;

(iv):

$\frac{2N-\alpha }{N}<p\le \frac{2N-2-\alpha }{N-2}$, $N\ge 5$ and $\lambda$ sufficiently large.

The arguments of this proof were adapted from articles [25, 30]. Observe that the conditions stated in this result are exactly the same of Theorem 1 and Theorem 13.

Proof

We know that $U(x)=\frac{[N(N-2)]^{\frac{N-2}{4}}}{(1+|x|^2)^{\frac{N-2}{2}}}$ is a minimizer for S, the best Sobolev constant of the immersion $D^{1,2}({\mathbb {R}}^N)\hookrightarrow L^{2^*}({\mathbb {R}}^N)$ (see [37, Theorem 1.42] or [13, Section 3]) and also a minimizer for $S_{H,L}$, according to Proposition 4.

If $B_r$ denotes the ball in ${\mathbb {R}}^N$ of center at origin and radius r, consider the balls $B_{\delta }$ and $B_{2\delta }$ and take $\psi \in C_0^{\infty }({\mathbb {R}}^N)$ such that, for a constant $C>0$,

$$\begin{aligned}\psi (x) = \left\{ \begin{array}{ll} 1, &{} \text {if}\;x \in B_{\delta },\\ 0, &{} \text {if}\;x \in {\mathbb {R}}^N\setminus B_{2\delta },\end{array}\right. \qquad 0\le |\psi (x)|\le 1,\ \ |D\psi (x)|\le C,\quad \forall \;x\in {\mathbb {R}}^N. \end{aligned}$$

We define, for $\varepsilon >0,$

$$\begin{aligned} U_\varepsilon (x):=\varepsilon ^{(2-N)/2}U\left( \displaystyle \frac{x}{\varepsilon }\right) \quad \text {and}\qquad u_\varepsilon (x):=\psi (x)U_\varepsilon (x). \end{aligned}$$

(21)

In the proof, we apply the estimates

$$\begin{aligned} \displaystyle \int \limits _{{\mathbb {R}}^N}|\nabla u_{\varepsilon }|^2 \text {d}x =C(N,\alpha )^{\frac{N-2}{2N-\alpha }\cdot \frac{N}{2}}S_{A}^{\frac{N}{2}}+O(\varepsilon ^{N-2}) \end{aligned}$$

(22)

and

$$\begin{aligned} \int \limits _{{\mathbb {R}}^N}\int \limits _{{\mathbb {R}}^N}\frac{|u_{\varepsilon }(x)|^{2_{\alpha }^*} |u_{\varepsilon }(y)|^{2_{\alpha }^*}}{|x-y|^{\alpha }}\text {d}x\text {d}y\ge C(N,\alpha )^{\frac{N}{2}}S_{A}^{\frac{2N-\alpha }{2}}-O(\varepsilon ^{N-\frac{\alpha }{2}}), \end{aligned}$$

(23)

which were obtained by Gao and Yang [26].

Case 1. $\frac{N+2-\alpha }{N-2}<p<2_{\alpha }^*$ and $N=3,4$ or $\frac{2N-2-\alpha }{N-2}<p<2_{\alpha }^*$ and $N\ge 5$.

Proof of Case 1

Consider the function $f:[0,+\infty )\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} f(t)=J_{A,V_{{\mathcal {P}}}}(tu_{\varepsilon })=\frac{t^2}{2}\Vert u_{\varepsilon }\Vert ^2_{A,V_{{\mathcal {P}}}}-\frac{t^{2\cdot 2_{\alpha }^*}}{2\cdot 2_{\alpha }^*}D(u_{\varepsilon })-\frac{\lambda t^{2p}}{2p}B(u_{\varepsilon }).\end{aligned}$$

The mountain pass geometry (Lemma 7) implies the existence of $t_{\varepsilon }>0$ such that $\displaystyle \sup _{t\ge 0} J_{A,V_{{\mathcal {P}}}} (t u_{\varepsilon })=J_{A,V_{{\mathcal {P}}}} (t_{\varepsilon } u_{\varepsilon })$. Since $t_{\varepsilon }>0$, $B(u_{\varepsilon })>0$ and $f'(t_{\varepsilon })=0$, we obtain

$$\begin{aligned} 0<t_{\varepsilon }<\left( \frac{\Vert u_{\varepsilon }\Vert _{A,V_{{\mathcal {P}}}}^2}{D(u_{\varepsilon })}\right) ^{\frac{1}{2(2_{\alpha }^*-1)}}:=S_{A}(\varepsilon ),\end{aligned}$$

thus implying

$$\begin{aligned} \Vert u_{\varepsilon }\Vert _{A,V_{{\mathcal {P}}}}^2= D(u_{\varepsilon })\left( S_{A}(\varepsilon )\right) ^{2(2_{\alpha }^*-1)}. \end{aligned}$$

(24)

Now define $g:[0,S_{A}(\varepsilon )]\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} g(t)=\frac{t^2}{2}\Vert u_{\varepsilon }\Vert ^2_{A,V_{{\mathcal {P}}}}-\frac{t^{2\cdot 2_{\alpha }^* }}{2\cdot 2_{\alpha }^*}D(u_\varepsilon ).\end{aligned}$$

So,

$$\begin{aligned} g(t)=\frac{t^2}{2}D(u_{\varepsilon })\left( S_{A}(\varepsilon )\right) ^{2(2_{\alpha }^*-1)}-\frac{t^{2\cdot 2_{\alpha }^* }}{2\cdot 2_{\alpha }^*}D(u_{\varepsilon }).\end{aligned}$$

Since $t>0$ and $D(u_{\varepsilon })>0$, it follows that $g'(t)>0$, and consequently, g is increasing in this interval. Thus,

$$\begin{aligned} 0<g(t_{\varepsilon })<\frac{N+2-\alpha }{2(2N-\alpha )}D(u_{\varepsilon })(S_{A}(\varepsilon ))^{2\cdot 2_{\alpha }^*}. \end{aligned}$$

We conclude that

$$\begin{aligned} D(u_{\varepsilon })(S_{A}(\varepsilon ))^{2\cdot 2_{\alpha }^*}=\frac{(\Vert u_{\varepsilon }\Vert _{A,V_{{\mathcal {P}}}}^2)^{\frac{2N-\alpha }{N+2-\alpha }}}{D(u_{\varepsilon })^{\frac{N-2}{N+2-\alpha }}} \end{aligned}$$

and therefore

$$\begin{aligned} 0<g(t_{\varepsilon })<\frac{N+2-\alpha }{2(2N-\alpha )}\cdot \frac{(\Vert u_{\varepsilon }\Vert _{A,V_{{\mathcal {P}}}}^2)^{\frac{2N-\alpha }{N+2-\alpha }}}{D(u_{\varepsilon })^{\frac{N-2}{N+2-\alpha }}}. \end{aligned}$$

Since $J_{A,V_{{\mathcal {P}}}}(t u_{\varepsilon })=g(t)-\frac{\lambda }{2p}t^{2p}B(u_{\varepsilon })$, we have

$$\begin{aligned} J_{A,V_{{\mathcal {P}}}}(t_{\varepsilon } u_{\varepsilon })< \frac{N+2-\alpha }{2(2N-\alpha )}\left( \frac{\Vert u_{\varepsilon }\Vert _{A,V_{{\mathcal {P}}}}^2}{D(u_{\varepsilon })^{\frac{N-2}{2N-\alpha }}}\right) ^{\frac{2N-\alpha }{N+2-\alpha }}-\frac{\lambda }{2p}t_{\varepsilon }^{2p}B(u_{\varepsilon }). \end{aligned}$$

But $\Vert u_{\varepsilon }\Vert _{A,V_{{\mathcal {P}}}}^2=\int \limits _{{\mathbb {R}}^N}|\nabla u_{\varepsilon }|^2 \text {d}x+\int \limits _{{\mathbb {R}}^N}(|A(x)|^2+V_{{\mathcal {P}}}(x)|u_{\varepsilon }|^2) \text {d}x$ implies

$$\begin{aligned} \frac{\Vert u_{\varepsilon }\Vert _{A,V_{{\mathcal {P}}}}^2}{D(u_{\varepsilon })^{\frac{N-2}{2N-\alpha }}}&=\frac{1}{(D(u_{\varepsilon }))^{\frac{N-2}{2N-\alpha }}}\int \limits _{{\mathbb {R}}^N}|\nabla u_{\varepsilon }|^2 \text {d}x +\frac{1}{(D(u_{\varepsilon }))^{\frac{N-2}{2N-\alpha }}}\int \limits _{{\mathbb {R}}^N}(|A(x)|^2+V_{{\mathcal {P}}}(x)|u_{\varepsilon }|^2 )\text {d}x. \end{aligned}$$

Therefore, we conclude that

$$\begin{aligned} J_{A,V_{{\mathcal {P}}}}(t_{\varepsilon }u_{\varepsilon })&<\frac{N+2-\alpha }{2(2N-\alpha )}\left( \frac{1}{(D(u_{\varepsilon }))^{\frac{N-2}{2N-\alpha }}}\displaystyle \int \limits _{{\mathbb {R}}^N}|\nabla u_{\varepsilon }|^2 \text {d}x \right. \\&\quad \left. +\frac{1}{(D(u_{\varepsilon }))^{\frac{N-2}{2N-\alpha }}}\displaystyle \int \limits _{{\mathbb {R}}^N}(|A(x)|^2+V_{{\mathcal {P}}}(x))|u_{\varepsilon }|^2 \text {d}x\right) ^{\frac{2N-\alpha }{N+2-\alpha }}-\frac{\lambda }{2p}t_{\varepsilon }^{2p}B(u_{\varepsilon }). \end{aligned}$$

Since, for all $\beta \ge 1$ and any $a,b>0$ we have $(a+b)^{\beta }\le a^{\beta }+\beta (a+b)^{\beta -1} b$, considering

$$\begin{aligned}a=\frac{1}{(D(u_{\varepsilon }))^{\frac{N-2}{2N-\alpha }}}\displaystyle \int \limits _{{\mathbb {R}}^N}|\nabla u_{\varepsilon }|^2 \text {d}x,\quad b=\frac{1}{(D(u_{\varepsilon }))^{\frac{N-2}{2N-\alpha }}}\displaystyle \int \limits _{{\mathbb {R}}^N}(|A(x)|^2+V_{{\mathcal {P}}}(x)|u_{\varepsilon }|^2 )\text {d}x\quad \text {and}\quad \beta =\frac{2N-\alpha }{N+2-\alpha }, \end{aligned}$$

it follows

$$\begin{aligned} J_{A,V_{{\mathcal {P}}}}(t_{\varepsilon }u_\varepsilon )&<\frac{N+2-\alpha }{2(2N-\alpha )}\left[ \left( \frac{1}{(D(u_{\varepsilon }))^{\frac{N-2}{2N-\alpha }}}\displaystyle \int \limits _{{\mathbb {R}}^N}|\nabla u_{\varepsilon }|^2 \text {d}x \right) ^{\frac{2N-\alpha }{N+2-\alpha }}\right. \\&\quad +\frac{2N-\alpha }{N+2-\alpha }\left( \frac{1}{D(u_{\varepsilon })^{\frac{N-2}{2N-\alpha }}}\displaystyle \int \limits _{{\mathbb {R}}^N} |\nabla u_{\varepsilon }|^2 \text {d}x + \frac{1}{(D(u_{\varepsilon })^{\frac{N-2}{2N-\alpha }}}\displaystyle \int \limits _{{\mathbb {R}}^N}(|A(x)|^2+V_{{\mathcal {P}}}(x)|u_{\varepsilon }|^2 )\text {d}x\right) ^{\frac{N-2}{N+2-\alpha }}\nonumber \\&\quad \cdot \left. \frac{1}{((D(u_{\varepsilon }))^{\frac{N-2}{2N-\alpha }}}\displaystyle \int \limits _{{\mathbb {R}}^N}(|A(x)|^2+V_{{\mathcal {P}}}(x)|u_{\varepsilon }|^2 )\text {d}x\right] -\frac{\lambda }{2p}t_{\varepsilon }^{2p}B(u_{\varepsilon })\nonumber . \end{aligned}$$

(25)

Taking into account (22) and (23), we conclude that

$$\begin{aligned} \left( \frac{1}{(D(u_{\varepsilon }))^{\frac{N-2}{2N-\alpha }}}\displaystyle \int \limits _{{\mathbb {R}}^N}|\nabla u_{\varepsilon }|^2 \text {d}x\right) ^{\frac{2N-\alpha }{N+2-\alpha }}&\le \left( \frac{(C(N,\alpha ))^{\frac{N-2}{2N-\alpha }\cdot \frac{N}{2}}\cdot S_{H,L}^{\frac{N}{2}}+O(\varepsilon ^{N-2})}{\left( C(N,\alpha )^{\frac{N}{2}}S_{H,L}^{\frac{2N-\alpha }{2}}-O(\varepsilon ^{\frac{2N-\alpha }{2}})\right) ^{\frac{N-2}{2N-\alpha }}}\right) ^{\frac{2N-\alpha }{N+2-\alpha }}. \end{aligned}$$

We also have

$$\begin{aligned} \left( \frac{(C(N,\alpha ))^{\frac{N-2}{2N-\alpha }\cdot \frac{N}{2}}(S_{H,L})^{\frac{N}{2}}+O(\varepsilon ^{N-2})}{\left( C(N,\alpha )^{\frac{N}{2}}S_{H,L}^{\frac{2N-\alpha }{2}}- O(\varepsilon ^{\frac{2N-\alpha }{2}})\right) ^{\frac{N-2}{2N-\alpha }}}\right) ^{\frac{2N-\alpha }{N+2-\alpha }}=(S_{H,L})^{\frac{2N-\alpha }{N+2-\alpha }}\cdot \left( \frac{1+O(\varepsilon ^{N-2})}{\left( 1-O\left( \varepsilon ^{\frac{2N-\alpha }{2}}\right) \right) ^{\frac{N-2}{2N-\alpha }}}\right) ^{\frac{2N-\alpha }{N+2-\alpha }} \end{aligned}$$

and

$$\begin{aligned} \left( \frac{1+O(\varepsilon ^{N-2})}{\left( 1-O\left( \varepsilon ^{\frac{2N-\alpha }{2}}\right) \right) ^{\frac{N-2}{2N-\alpha }}}\right) ^{\frac{2N-\alpha }{N+2-\alpha }} < 1+C(N,\alpha )\cdot \frac{O(\varepsilon ^{N-2})+O(\varepsilon ^{\frac{2N-\alpha }{2}})}{\left( 1-O(\varepsilon ^{\frac{2N-\alpha }{2}})\right) ^{\frac{N-2}{2N-\alpha }}}. \end{aligned}$$

We observe that for $\varepsilon >0$ sufficiently small, it holds

$$\begin{aligned}(1-O(\varepsilon ^{\frac{N-2}{2N-\alpha }}))^{\frac{N-2}{2N-\alpha }}\ge \frac{1}{2}.\end{aligned}$$

So,

$$\begin{aligned} \left( \frac{1+O(\varepsilon ^{N-2})}{\left( 1-O\left( \varepsilon ^{\frac{2N-\alpha }{2}}\right) \right) ^{\frac{N-2}{2N-\alpha }}}\right) ^{\frac{2N-\alpha }{N+2-\alpha }}<1+2C(N,\alpha )\left( O\left( \varepsilon ^{N-2}\right) +O\left( \varepsilon ^{\frac{2N-\alpha }{2}}\right) \right) < 1+ O\left( \varepsilon ^{\min \{N-2,\frac{2N-\alpha }{2}\}}\right) . \end{aligned}$$

Therefore, we conclude that for any $\varepsilon >0$ sufficiently small, we have

$$\begin{aligned} \left( \frac{1}{(D(u_{\varepsilon }))^{\frac{N-2}{2N-\alpha }}}\displaystyle \int \limits _{{\mathbb {R}}^N}|\nabla u_{\varepsilon }|^2 \text {d}x\right) ^{\frac{2N-\alpha }{N+2-\alpha }} < \left( S_{H,L}\right) ^{\frac{2N-\alpha }{N+2-\alpha }}+ O\left( \varepsilon ^{\min \{N-2,\frac{2N-\alpha }{2}\}}\right) . \end{aligned}$$

(26)

Combining (25) with (26), for $\varepsilon$ sufficiently small, we have

$$\begin{aligned} J_{A,V_{{\mathcal {P}}}}(t_{\varepsilon }u_{\varepsilon })&<\frac{N+2-\alpha }{2(2N-\alpha )}\left( S_{H,L} \right) ^{\frac{2N-\alpha }{N+2-\alpha }}+ O\left( \varepsilon ^{\min \{N-2,\frac{2N-\alpha }{2}\}}\right) \\&\quad +\frac{1}{2}\left( \frac{1}{D(u_{\varepsilon })^{\frac{N-2}{2N-\alpha }}}\int \limits _{{\mathbb {R}}^N} |\nabla u_{\varepsilon }|^2 \text {d}x + \frac{1}{D(u_{\varepsilon })^{\frac{N-2}{2N-\alpha }}} \int \limits _{{\mathbb {R}}^N}(|A(x)|^2+V_{{\mathcal {P}}}(x))|u_{\varepsilon }|^2)\text {d}x\right) ^{\frac{N-2}{N+2-\alpha }} \nonumber \\&\quad \,\cdot \frac{1}{D(u_{\varepsilon })^{\frac{N-2}{2N-\alpha }}} \int \limits _{{\mathbb {R}}^N}(|A(x)|^2+V_{{\mathcal {P}}}(x))|u_{\varepsilon }|^2 \text {d}x-\frac{\lambda }{2p}t_{\varepsilon }^{2p}B(u_{\varepsilon }).\nonumber \end{aligned}$$

(27)

We claim that there is a positive constant $C_0$ such that for all $\varepsilon >0$,

$$\begin{aligned} t_{\varepsilon }^{2p}\ge C_0. \end{aligned}$$

(28)

In fact, suppose that there is a sequence $(\varepsilon _n)\subset {\mathbb {R}}$, $\varepsilon _n\rightarrow 0$ as $n\rightarrow \infty$, such that $t_{\varepsilon _n}\rightarrow 0$ as $n\rightarrow \infty$. Thus,

$$\begin{aligned}0<c_{\lambda }\le \sup _{t\ge 0}J_{A,V}(t u_{\varepsilon _n})=J_{A,V_{{\mathcal {P}}}}(t_{\varepsilon _n}u_{\varepsilon _n}).\end{aligned}$$

Since $u_{\varepsilon _n}\in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$ is bounded and $t_{\varepsilon _n}\rightarrow 0$, as $n\rightarrow \infty$, we have $t_{\varepsilon _n}u_{\varepsilon _n}\rightarrow 0$ as $n\rightarrow \infty$, in $H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$.

The continuity of $J_{A,V_{{\mathcal {P}}}}$ implies that $J_{A,V_{{\mathcal {P}}}}(t_{\varepsilon _n}u_{\varepsilon _n})\rightarrow J_{A,V_{{\mathcal {P}}}}(0)= 0$. Therefore,

$$\begin{aligned}0<c_{\lambda }\le \lim _{n\rightarrow \infty }J_{A,V_{{\mathcal {P}}}}(t_{\varepsilon _n}u_{\varepsilon _n})=0,\end{aligned}$$

a contradiction that proves the claim.

From (24), (27) and (28), we conclude that for some constants $C_0>0$ and $\varepsilon >0$ sufficiently small, we have

$$\begin{aligned} J_{A,V_{{\mathcal {P}}}}(t_{\varepsilon }u_{\varepsilon })&<\frac{N+2-\alpha }{2(2N-\alpha )}\left( S_{A} \right) ^{\frac{2N-\alpha }{N+2-\alpha }}+ O\left( \varepsilon ^{\min \{N-2,\frac{2N-\alpha }{2}\}}\right) \nonumber \\&\quad +\frac{1}{2}\left( \frac{1}{D(u_{\varepsilon })^{\frac{N-2}{2N-\alpha }}}\Vert u_{\varepsilon }\Vert ^2_{A V_{\mathcal {P}}}\right) ^{\frac{N-2}{N+2-\alpha }}\nonumber \cdot \frac{1}{\left( D(u_{\varepsilon }\right) ^{\frac{N-2}{2N-\alpha }}}\displaystyle \int \limits _{{\mathbb {R}}^N}(|A(x)|^2+V_{{\mathcal {P}}}(x))|u_{\varepsilon }|^2 \text {d}x-C_0B(u_{\varepsilon })\nonumber \\&<\frac{N+2-\alpha }{2(2N-\alpha )}\left( S_{A} \right) ^{\frac{2N-\alpha }{N+2-\alpha }}\\&\quad + O\left( \varepsilon ^{\min \{N-2,\frac{2N-\alpha }{2}\}}\right) +\frac{S_{A}(\varepsilon )^2}{2}\cdot \displaystyle \int \limits _{{\mathbb {R}}^N}(|A(x)|^2+V_{{\mathcal {P}}}(x))|u_{\varepsilon }|^2 \text {d}x-C_0B(u_{\varepsilon })\nonumber \\&=\frac{N+2-\alpha }{2(2N-\alpha )}\left( S_{A} \right) ^{\frac{2N-\alpha }{N+2-\alpha }}+ O(\varepsilon ^{\eta })+ C_1\displaystyle \int \limits _{{\mathbb {R}}^N} a(x)|u_{\varepsilon }|^2 \text {d}x-C_0B(u_{\varepsilon }),\nonumber \end{aligned}$$

(29)

where $C_1=\frac{S_{A}(\varepsilon )^2}{2}$, $a(x)=|A(x)|^2+V_p(x)$ and $\eta =\min \{N-2,\frac{2N-\alpha }{2}\}$.

By direct computation, we know that for $\varepsilon <1$, since $\psi (x)=0$ for all $x\in {\mathbb {R}}^N\setminus B_{2\delta }$ and $\psi \equiv 1$ in $B_{\delta }$, we have

$$\begin{aligned} B(u_{\varepsilon })&= \int \limits _{{\mathbb {R}}^N}\int \limits _{{\mathbb {R}}^N}\frac{|u_{\varepsilon }(x)|^p |u_{\varepsilon } (y)|^p}{|x-y|^{\alpha }}\text {d}x\text {d}y= \displaystyle \int \limits _{{\mathbb {R}}^N}\int \limits _{{\mathbb {R}}^N}\frac{|\psi (x) U_{\varepsilon }(x)|^p |\psi (y) U_{\varepsilon } (y)|^p}{|x-y|^{\alpha }}\text {d}x\text {d}y\\&= \int \limits _{B_{2\delta }}\int \limits _{B_{2\delta }}\frac{| \psi (x)U_{\varepsilon }(x)|^p |\psi (y) U_{\varepsilon } (y)|^p}{|x-y|^{\alpha }}\text {d}x\text {d}y \ge \displaystyle \int \limits _{B_{\delta }}\int \limits _{B_{\delta }}\frac{| U_{\varepsilon }(x)|^p | U_{\varepsilon } (y)|^p}{|x-y|^{\alpha }}\text {d}x\text {d}y\\&=\displaystyle \int \limits _{B_{\delta }}\displaystyle \int \limits _{B_{\delta }}\frac{\varepsilon ^{\frac{(2-N)p}{2}}[N(N-2)]^{\frac{(N-2)p}{4}}\varepsilon ^{\frac{(2-N)p}{2}}[N(N-2)]^{\frac{(N-2)p}{4}}}{(1+|\frac{x}{\varepsilon }|^2)^{\frac{(N-2)p}{2}}|x-y|^{\alpha }(1+|\frac{y}{\varepsilon }|^2)^{\frac{(N-2)p}{2}}}\text {d}x\text {d}y\\&\ge [N(N-2)]^{\frac{(N-2)p}{2}}\varepsilon ^{2N-\alpha -(N-2)p}\displaystyle \int \limits _{B_{\delta }}\displaystyle \int \limits _{B_{\delta }}\frac{1}{(1+|x|^2)^{\frac{(N-2)p}{2}}|x-y|^{\alpha }(1+|y|^2)^{\frac{(N-2)p}{2}}}\text {d}x\text {d}y\\&=C_3 \varepsilon ^{2N-\alpha -(N-2)p}. \end{aligned}$$

Since a(x) is bounded, (29) and the last inequality imply that

$$\begin{aligned} J_{A,V_{{\mathcal {P}}}}(t_{\varepsilon }u_{\varepsilon })< \frac{N+2-\alpha }{2(N-\alpha )}\left( S_{A} \right) ^{\frac{2N-\alpha }{N+2-\alpha }}+ O(\varepsilon ^{\eta })+ C_2\displaystyle \int \limits _{{\mathbb {R}}^N}|u_{\varepsilon }(x)|^2 \text {d}x-C_3 \varepsilon ^{2N-\alpha -(N-2)p}. \end{aligned}$$

(30)

We are going to see that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\varepsilon ^{-\eta }\left( C_2\displaystyle \int \limits _{{\mathbb {R}}^N}|u_{\varepsilon }(x)|^2 \text {d}x-C_3\varepsilon ^{2N-\alpha -(N-2)p}\right) =-\infty . \end{aligned}$$

(31)

In order to do that, it suffices to show that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\varepsilon ^{-\eta }\left( C_2\displaystyle \int \limits _{B_{\delta }}|u_{\varepsilon }(x)|^2 \text {d}x-C_3\varepsilon ^{2N-\alpha -(N-2)p}\right) =-\infty \end{aligned}$$

(32)

and

$$\begin{aligned} C_2\displaystyle \int \limits _{B_{2\delta }\setminus B_{\delta }}|u_{\varepsilon }(x)|^2 \text {d}x-C_3\varepsilon ^{2N-\alpha -(N-2)p}=O(\varepsilon ^{\eta }). \end{aligned}$$

(33)

Assuming (32) and (33), let us proceed with our proof. Since

$$\begin{aligned}O(\varepsilon ^{\eta }){+}C_2\displaystyle \int \limits _{{\mathbb {R}}^N}|u_{\varepsilon }(x)|^2 \text {d}x-C_3\varepsilon ^{2N-\alpha -(N-2)p}{=}\varepsilon ^{\eta }\left[ \frac{O(\varepsilon ^{\eta })}{\varepsilon ^\eta }+\varepsilon ^{-\eta }\left( C_2\displaystyle \int \limits _{{\mathbb {R}}^N}|u_{\varepsilon }(x)|^2 \text {d}x-C_3\varepsilon ^{2N-\alpha -(N-2)p}\right) \right] ,\end{aligned}$$

from (31) follows

$$\begin{aligned} O(\varepsilon ^{\eta })+C_2\displaystyle \int \limits _{{\mathbb {R}}^N}|u_{\varepsilon }(x)|^2 \text {d}x-C_3\varepsilon ^{2N-\alpha -(N-2)p}<0 \end{aligned}$$

(34)

for $\varepsilon >0$ sufficiently small.

Thus, (30) and (34) imply

$$\begin{aligned} \sup _{t\ge 0} J_{A,V_{{\mathcal {P}}}}(t u_{\varepsilon }) <\frac{N+2-\alpha }{2(2N-\alpha )}\left( S_{A} \right) ^{\frac{2N-\alpha }{N+2-\alpha }} \end{aligned}$$

for $\varepsilon >0$ sufficiently small and fixed. Once (32) and (33) are verified, the proof of Case 1 is complete. $\square$

We now prove (32).

Lemma 15

If $\frac{N+2-\alpha }{N-2}<p<2_{\alpha }^*$ and $N=3,4$ or $\frac{2N-2-\alpha }{N-2}<p<2_{\alpha }^*$ and $N\ge 5$, it follows that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\varepsilon ^{-\eta }\left( C_2\displaystyle \int \limits _{B_{\delta }}|u_{\varepsilon }(x)|^2 \text {d}x-C_3\varepsilon ^{2N-\alpha -(N-2)p}\right) =-\infty . \end{aligned}$$

Proof

This limit is evaluated considering the cases $N=3,$$N=4$ and $N\ge 5$ as follows. We initially observe that direct computation allows us to conclude that

$$\begin{aligned} \displaystyle \int \limits _{B_{\delta }}|u_{\varepsilon }(x)|^2 \text {d}x= N\omega _N [N(N-2)]^{\frac{N-2}{2}}\varepsilon ^2\displaystyle \int \limits _{0}^{\frac{\delta }{\varepsilon }}\frac{r^{N-1}}{(1+r^2)^{N-2}} \text {d}r, \end{aligned}$$

(35)

where $\omega _N$ denotes the volume of the unit ball in ${\mathbb {R}}^N$.

Now, define

$$\begin{aligned} I_{\varepsilon }:&=\varepsilon ^{-\eta }\left( C_2\displaystyle \int \limits _{B_{\delta }}|u_{\varepsilon }(x)|^2 \text {d}x-C_3\varepsilon ^{2N-\alpha -(N-2)p}\right) =\varepsilon ^{-\eta }\left( C_4\varepsilon ^2\displaystyle \int \limits _{0}^{\frac{\delta }{\varepsilon }}\frac{r^{N-1}}{(1+r^2)^{N-2}} \text {d}r-C_3\varepsilon ^{2N-\alpha -(N-2)p}\right) , \end{aligned}$$

the second equality being a consequence of (35).

$\bullet$The case $\mathbf {N=3}$. In this case, we have $5-\alpha<p<2^*_\alpha$ and therefore $5-\alpha -p<0$. We also observe that $0<\alpha <N$ implies $\min \{N-2,\frac{2N-\alpha }{2}\}=N-2=1$.

It is easy to show that

$$\begin{aligned} \varepsilon ^2\displaystyle \int \limits _{0}^{\frac{\delta }{\varepsilon }}\frac{r^2}{1+r^2} \text {d}r=\varepsilon \left( \delta -\varepsilon \arctan \left( \frac{\delta }{\varepsilon }\right) \right) . \end{aligned}$$

Thus,

$$\begin{aligned} I_{\varepsilon }&=C_4\left( \delta -\varepsilon \arctan \left( \frac{\delta }{\varepsilon }\right) \right) -C_3\varepsilon ^{5-\alpha -p}. \end{aligned}$$

Our claim follows.

$\bullet$The case $\mathbf {N= 4}$. In this case, $\frac{6-\alpha }{2}<p<2_{\alpha }^*$ implies $6-\alpha -2p<0$ and $\min \{N-2,\frac{2N-\alpha }{2}\}=N-2=2$, since $0<\alpha <4$.

We have

$$\begin{aligned} \varepsilon ^2\displaystyle \int \limits _{0}^{\frac{\delta }{\varepsilon }}\frac{r^3}{(1+r^2)^2} \text {d}r=\frac{\varepsilon ^2}{2}\left[ \ln \left( 1+\frac{\delta ^2}{\varepsilon ^2}\right) + \frac{\varepsilon ^2}{\varepsilon ^2+\delta ^2}-1\right] . \end{aligned}$$

So,

$$\begin{aligned} I_{\varepsilon }&=\frac{C_4}{2} \left( \ln \left( 1+\frac{\delta ^2}{\varepsilon ^2}\right) + \frac{\varepsilon ^2}{\varepsilon ^2+\delta ^2}-1\right) -C_3\varepsilon ^{6-\alpha -2p}.\\ \end{aligned}$$

Our claim follows.

$\bullet$The case $\mathbf {N\ge 5}$. We have

$$\begin{aligned} I_{\varepsilon }&=\varepsilon ^{2-\min \{N-2,\frac{2N-\alpha }{2}\}}\left( C_4\displaystyle \int \limits _{0}^{\frac{\delta }{\varepsilon }}\frac{r^{N-1}}{(1+r^2)^{N-2}} \text {d}r-C_3\varepsilon ^{2N-\alpha -(N-2)p-2}\right) . \end{aligned}$$

It is easy to show that if $N\ge 5$, then the integral

$$\begin{aligned}\lim _{\varepsilon \rightarrow 0}\int \limits _{0}^{\frac{\delta }{\varepsilon }}\frac{r^{N-1}}{(1+r^2)^{N-2}} \text {d}r\end{aligned}$$

converges.

There are two cases to be considered:

• $0<\alpha <4$ and $N\ge 5$;

• $\alpha \ge 4$ and $N\ge 5$.

Let us suppose $0<\alpha <4$ and $N\ge 5$. Since $0<\alpha <4$, we have

$$\begin{aligned}2-\eta =2-\min \{N-2, \frac{2N-\alpha }{2}\}=-N+4<0.\end{aligned}$$

Also $\frac{2N-\alpha -2}{N-2}<p<\frac{2N-\alpha }{N-2}$ implies $2N-\alpha -(N-2)p-2<0$. Therefore, $I_{\varepsilon }\rightarrow -\infty$ as $\varepsilon \rightarrow 0$.

Now we consider the case $\alpha \ge 4$ and $N\ge 5$. We have $N-2\ge \frac{2N-\alpha }{2}$ and therefore

$$\begin{aligned}2-\eta =2-\min \bigg \{N-2, \frac{2N-\alpha }{2}\bigg \}=2-N+\frac{\alpha }{2}<0.\end{aligned}$$

Since

$$\begin{aligned}I_{\varepsilon }=\varepsilon ^{2-N+\frac{\alpha }{2}}\left[ C_4\int \limits _{0}^{\frac{\delta }{\varepsilon }}\frac{r^{N-1}}{(1+r^2)^{N-2}} \text {d}r-C_3\varepsilon ^{2N-\alpha -(N-2)p-2}\right] ,\end{aligned}$$

we conclude that $I_{\varepsilon }\rightarrow -\infty$. We are done. $\square$

We now prove (33).

Lemma 16

It holds

$$\begin{aligned} C_2\displaystyle \int \limits _{B_{2\delta }\setminus B_{\delta }}|u_{\varepsilon }(x)|^2 \text {d}x-C_3\varepsilon ^{2N-\alpha -(N-2)p}=O(\varepsilon ^{\eta }). \end{aligned}$$

Proof

Fix $\delta >0$ sufficiently large so that $U^2_{\varepsilon } (x)\le \varepsilon ^{1+\eta }$ if $|x|\ge \delta$. Since

$$\begin{aligned} \frac{1}{\varepsilon ^{\eta }}\left[ C_2\displaystyle \int \limits _{B_{2\delta }\setminus B_{\delta }}|u_{\varepsilon }(x)|^2 \text {d}x-C_3\varepsilon ^{2N-\alpha -(N-2)p}\right]&<\frac{C_2}{\varepsilon ^{\eta }}\displaystyle \int \limits _{B_{2\delta }\setminus B_{\delta }} \psi ^2 (x) U^2_{\varepsilon }(x)\text {d}x\le C_2\varepsilon \Vert \psi \Vert _{2}\\&\le C_1\varepsilon \Vert \psi \Vert _{A,V_{{\mathcal {P}}}}, \end{aligned}$$

our proof is complete. $\square$

Case 2. For $\lambda$ sufficiently large, $\frac{2N-\alpha }{N}<p\le \frac{N+2-\alpha }{N-2}$ and $N=3,4$ or $\frac{2N-\alpha }{N}<p\le \frac{2N-2-\alpha }{N-2}$ and $N\ge 5.$

Proof of Case 2

Define $g_\lambda :[0,+\infty )\rightarrow {\mathbb {R}}$ by

$$\begin{aligned}g_{\lambda } (t)= J_{A,V_{{\mathcal {P}}}}(t u_{\varepsilon })= \frac{t^2}{2}\int \limits _{{\mathbb {R}}^N} \left[ |\nabla u_{\varepsilon }|^2+\left( |A(x)|^2+V_{{\mathcal {P}}}(x)\right) |u_{\varepsilon }|^2\right] \text {d}x-\frac{\lambda }{2p}t^{2p}B(u_{\varepsilon })-\frac{1}{2\cdot 2_{\alpha }^*}t^{2\cdot 2_{\alpha }^*} D(u_{\varepsilon }).\end{aligned}$$

We already know that $\displaystyle \max _{t\ge 0} g_{\lambda } (t)$ is attained at some $t_{\lambda }>0.$ Since $g'_{\lambda } (t_\lambda )=0$, we have

$$\begin{aligned}\int \limits _{{\mathbb {R}}^N} \left[ |\nabla u_{\varepsilon }|^2+\left( |A(x)|^2+V_{{\mathcal {P}}}(x)\right) |u_{\varepsilon }|^2\right] \text {d}x= \lambda t_{\lambda }^{2(p-1)}B(u_{\varepsilon })+t_{\lambda }^{2( 2_{\alpha }^*-1)} D(u_{\varepsilon }).\end{aligned}$$

Thus, $t_{\lambda }\rightarrow 0$ as $\lambda \rightarrow +\infty$ and

$$\begin{aligned} \max _{t\ge 0} J_{A,V_{{\mathcal {P}}}}(t u_{\varepsilon })&= \frac{{t_\lambda }^2}{2}\int \limits _{{\mathbb {R}}^N} \left[ |\nabla u_{\varepsilon }(x)|^2+\left( |A(x)|^2+V_{{\mathcal {P}}}(x)\right) |u_{\varepsilon }(x)|^2\right] \text {d}x-\frac{\lambda }{2p}{t_\lambda }^{2p}B(u_{\varepsilon })-\frac{1}{2\cdot 2_{\alpha }^*}t^{2\cdot 2_{\alpha }^*} D(u_{\varepsilon })\\&< \frac{{t_\lambda }^2}{2}\int \limits _{{\mathbb {R}}^N} \left[ |\nabla u_{\varepsilon }|^2+\left( |A(x)|^2+V_{{\mathcal {P}}}(x)\right) |u_{\varepsilon }(x)|^2\right] \text {d}x. \end{aligned}$$

Since $t_{\lambda }\rightarrow 0$ as $\lambda \rightarrow +\infty$ and $\frac{N+2-\alpha }{2(N-\alpha )}(S_{A})^{\frac{2N-\alpha }{N+2-\alpha }}>0$, we conclude that

$$\begin{aligned}\frac{{t_\lambda }^2}{2}\int \limits _{{\mathbb {R}}^N} \left[ |\nabla u_{\varepsilon }|^2+\left( |A(x)|^2+V_{{\mathcal {P}}}(x)\right) |u_{\varepsilon }(x)|^2\right] \text {d}x<\frac{N+2-\alpha }{2(2N-\alpha )}(S_{A})^{\frac{2N-\alpha }{N+2-\alpha }},\end{aligned}$$

for $\lambda >0$ sufficiently large.

Therefore,

$$\begin{aligned}\sup _{t\ge 0}J_{A,V_{{\mathcal {P}}}}(t u_{\varepsilon })<\frac{N+2-\alpha }{2(2N-\alpha )}(S_{A})^{\frac{2N-\alpha }{N+2-\alpha }}\end{aligned}$$

for $\lambda >0$ sufficiently large. $\square$

3.2 The proof of Theorem 1

Some arguments of this proof were adapted from articles [2, 30].

Maintaining the notation introduced in subsection 3.1, consider the energy functional $I_{A,V}:H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})\rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} I_{A,V}(u)=\frac{1}{2}{\Vert u\Vert ^2}_{A,V} -\frac{1}{2\cdot 2_{\alpha }^*}D(u)-\frac{\lambda }{2p}B(u). \end{aligned}$$

We denote by ${\mathcal {N}}_{A,V}$ the Nehari Manifold related to $I_{A,V}$, that is,

$$\begin{aligned}{\mathcal {N}}_{A,V}&=\left\{ u\in H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})\setminus \{0\}\,:\,{\Vert u\Vert }^2_{A,V}=D(u)+\lambda B(u)\right\} , \end{aligned}$$

which is nonempty as a consequence of Theorem 13. As before, the functional $I_{A,V}$ satisfies the mountain pass geometry. Thus, there exists a sequence $(u_n)\subset H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})$ such that

$$\begin{aligned}I'_{A,V}(u_n)\rightarrow 0\qquad \text {and}\qquad I_{A,V}(u_n)\rightarrow d_{\lambda },\end{aligned}$$

where $d_{\lambda }$ is the minimax level, also characterized by

$$\begin{aligned} d_{\lambda }=\inf _{u\;\in H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})\setminus \{0\}}\max _{t\ge 0}I_{A,V}(tu)=\inf _{{\mathcal {N}}_{A,V}} I_{A,V}(u)>0. \end{aligned}$$

We stress that as a consequence of ($V_2$), we have $I_{A,V}(u)<J_{A,V_{{\mathcal {P}}}}(u)$ for all $u\in H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})$.

The next lemma compares the levels $d_{\lambda }$ and $c_{\lambda }$.

Lemma 17

The levels $d_{\lambda }$ and $c_{\lambda }$ verify the inequality

$$\begin{aligned}d_{\lambda }<c_{\lambda } <\frac{N+2-\alpha }{2(2N-\alpha )}(S_{A})^{\frac{2N-\alpha }{N+2-\alpha }}\end{aligned}$$

for all $\lambda >0$.

Proof

Let u be the ground-state solution of problem (9) and consider ${\bar{t}}_u>0$ such that ${\bar{t}}_u u \in {\mathcal {N}}_{A,V}$, that is,

$$\begin{aligned} 0<d_{\lambda }\le \sup _{t\ge 0} I_{A,V}(tu)=I_{A,V}({\bar{t}}_u u). \end{aligned}$$

It follows from $(V_2)$ that

$$\begin{aligned}0<d_{\lambda }\le I_{A,V}({\bar{t}}_u u)<J_{A,V_{{\mathcal {P}}}}({\bar{t}}_u u)\le \sup _{t\ge 0} J_{A,V_{{\mathcal {P}}}}(tu)= J_{A,V_{{\mathcal {P}}}}(u)=c_{\lambda }.\end{aligned}$$

Therefore,

$$\begin{aligned}d_\lambda <c_{\lambda }.\end{aligned}$$

The second inequality was already known. $\square$

Proof of Theorem 1

Let $(u_n)$ be a $(PS)_{d_{\lambda }}$ sequence for $I_{A,V}$. As before, $(u_n)$ is bounded in $H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})$. Thus, there exists $u\in H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})$ such that

$$\begin{aligned}u_n\rightharpoonup u\ \ \text {in}\ \ H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}}).\end{aligned}$$

By the same arguments given in the proof of Theorem 13, u is a ground-state solution of problem (4), if $u\ne 0$.

Following close [2], we will show that $u=0$ cannot occur. Indeed, Lemma  6 yields

$$\begin{aligned}\lim _{n\rightarrow \infty }\int \limits _{{\mathbb {R}}^N} W|u_n|^2\;\text {d}x=0, \end{aligned}$$

since $W \in L^{\frac{N}{2}}({\mathbb {R}}^N,{\mathbb {C}})$ and $u_n\rightharpoonup 0$ in $H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})$. So,

$$\begin{aligned}|J_{A,V_{{\mathcal {P}}}}(u_n)-I_{A,V}(u_n)|=o_n (1),\end{aligned}$$

showing that

$$\begin{aligned}J_{A,V_{{\mathcal {P}}}}(u_n)\rightarrow d_{\lambda }.\end{aligned}$$

But, for $\varphi \in H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})$ such that $\Vert \varphi \Vert _{A,V}\le 1$, we have

$$\begin{aligned} |(J'_{A,V_{{\mathcal {P}}}}(u_n)-I'_{A,V}(u_n))\cdot \varphi |\le \bigg (\int \limits _{{\mathbb {R}}^N} W|u_n|^2\; \text {d}x\bigg )^{\frac{1}{2}}=o_n(1). \end{aligned}$$

Thus,

$$\begin{aligned}J'_{A,V_{{\mathcal {P}}}}(u_n)=o_n (1)\end{aligned}$$

Let $t_n>0$ such that $t_n u_n \in {\mathcal {M}}_{A,V_{{\mathcal {P}}}}$. Mimicking the argument found in [1, 23, 35, 37], it follows that $t_n\rightarrow 1$ as $n\rightarrow \infty$. Therefore,

$$\begin{aligned}c_{\lambda }\le J_{A,V_{{\mathcal {P}}}}(t_n u_n)=J_{A,V_{{\mathcal {P}}}}(u_n)+o_n (1)=d_{\lambda }+o_n(1).\end{aligned}$$

Letting $n\rightarrow +\infty$, we get

$$\begin{aligned}c_{\lambda }\le d_{\lambda }\end{aligned}$$

obtaining a contradiction with Lemma 17. This completes the proof of Theorem 1. $\square$

4 The case $f(u)=|u|^{p-1} u$

4.1 The periodic problem

In this subsection, we deal with problem (5) for f(u) as above, that is,

$$\begin{aligned} -(\nabla +iA(x))^2u+ V_{{\mathcal {P}}}(x)u =\left( \frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\right) |u|^{2_{\alpha }^*-2} u + \lambda |u|^{p-1} u, \end{aligned}$$

(36)

where $1<p<2^{*} - 1$.

We observe that in this case, the energy functional $J_{A,V_{{\mathcal {P}}}}$ is given by

$$\begin{aligned} J_{A,V_{{\mathcal {P}}}}(u):=\frac{1}{2}\Vert u\Vert ^2_{A,V_{{\mathcal {P}}}}-\frac{1}{2\cdot 2_{\alpha }^*}D(u)-\frac{\lambda }{p+1}\int \limits _{{\mathbb {R}}^N}|u|^{p+1} \text {d}x, \end{aligned}$$

where, as before,

$$\begin{aligned}D(u)=\int \limits _{{\mathbb {R}}^N}\left( \frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\right) |u|^{2_{\alpha }^*}\,\text {d}x=\int \limits _{{\mathbb {R}}^N}\int \limits _{{\mathbb {R}}^N}\frac{|u(x)^{2_{\alpha }^*}| |u(y)|^{2_{\alpha }^*}}{|x-y|^{\alpha }}\text {d}x\text {d}y.\end{aligned}$$

By the Sobolev immersion (6) and the Hardy–Littlewood–Sobolev inequality, we have that $J_{A,V_{{\mathcal {P}}}}$ is well defined.

Definition 4.1

A function $u\in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$ is a weak solution of (36) if

$$\begin{aligned}\langle u,\varphi \rangle _{A,V_{{\mathcal {P}}}} -\mathfrak {Re}\int \limits _{{\mathbb {R}}^N}\left( \frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\right) |u|^{2_{\alpha }^*-2}u{\bar{\psi }}\,\text {d}x-\lambda \;\mathfrak {Re}\int \limits _{{\mathbb {R}}^N} |u|^{p-1}u{\bar{\psi }}\,\text {d}x =0\end{aligned}$$

for all $\psi \in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$.

As before, we see that critical points of $J_{A,V_{{\mathcal {P}}}}$ are weak solutions of (36) and

$$\begin{aligned} J'_{A,V_{{\mathcal {P}}}}(u)\cdot u:=\Vert u\Vert _{A,V_{{\mathcal {P}}}}^{2}-D(u)-\lambda \Vert u\Vert ^{p+1}_{p+1}. \end{aligned}$$

We obtain that $J_{A,V_{{\mathcal {P}}}}$ satisfies the geometry of the mountain pass (see the proof of Lemma 7).

As in Section 3, the mountain pass theorem without the PS condition yields a sequence $(u_n)\subset H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})$ such that

$$\begin{aligned}J'_{A,V_{{\mathcal {P}}}}(u_n)\rightarrow 0\qquad \text {and}\qquad J_{A,V_{{\mathcal {P}}}}(u_n)\rightarrow c_{\lambda },\end{aligned}$$

where $c_{\lambda }=\inf _{\alpha \in \Gamma }\max _{t\in [0,1]}J_{A,V_{{\mathcal {P}}}}(\gamma (t))$ and $\Gamma {=}\left\{ \gamma \in C^1\left( [0,1],H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})\right) \,:\,\gamma (0){=}0,\, J_{A,V_{{\mathcal {P}}}}(\gamma (1)){<}0\right\}$.

Considering the Nehari manifold associated with $J_{A,V_{{\mathcal {P}}}}$, that is,

$$\begin{aligned}{\mathcal {M}}_{A,V_{{\mathcal {P}}}}&=\left\{ u\in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})\setminus \{0\}\,:\,\Vert u\Vert _{A,V_{{\mathcal {P}}}}^{2}= D(u) + \lambda \Vert u\Vert ^{p+1}_{p+1}\right\} , \end{aligned}$$

and proceeding as in the proof of Lemma 11 we obtain.

Lemma 18

There exists a unique $t_u=t_u (u)>0$ such that $t_u u\in {\mathcal {M}}_{A,V_{{\mathcal {P}}}}$ for all $u\in H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})\setminus \{0\}$ and $J_{A,V_{{\mathcal {P}}}}(t_u u)=\displaystyle \max _{t\ge 0} J_{A,V_{{\mathcal {P}}}}(tu)$. Moreover, $c_{\lambda }=c^*_{\lambda }=c^{**}_{\lambda }$, where

$$\begin{aligned}c_{\lambda }^{*}= \displaystyle \inf _{u \;\in \; {\mathcal {M}}_{A,V_{{\mathcal {P}}}}} J_{A,V_{{\mathcal {P}}}}(u)\quad \text {and}\quad c_{\lambda }^{**}=\displaystyle \inf _{u\;\in \; H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})\setminus \{0\}}\max _{t\ge 0} J_{A,V_{{\mathcal {P}}}}(tu).\end{aligned}$$

Lemma 19

Suppose that $u_n\rightharpoonup u_0$ and consider

$$\begin{aligned} B'(u_n)\cdot \psi =\mathfrak {Re}\int \limits _{{\mathbb {R}}^N}|u|^{p-1}u{\bar{\psi }}\end{aligned}$$

and

$$\begin{aligned} D'(u_n)\cdot \psi =\mathfrak {Re}\int \limits _{{\mathbb {R}}^N}\bigg (\frac{1}{|x|^{\alpha }}*|u_n|^{2_{\alpha }^*}\bigg )|u_n|^{2_{\alpha }^*-2}u_n{\bar{\psi }}\end{aligned}$$

for $\psi \in C^\infty _c({\mathbb {R}}^{N},{\mathbb {C}})$. Then, $B'(u_n)\cdot \psi \rightarrow B'(u_0)\cdot \psi$ and $D'(u_n)\cdot \psi \rightarrow D'(u_0)\cdot \psi$ as $n\rightarrow \infty .$

Lemma 20

If $(u_n)\subset H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})$ is a Palais–Smale sequence for $J_{A,V_{{\mathcal {P}}}}$, then $(u_n)$ is bounded. In addition, if $u_n\rightharpoonup u$ weakly in $H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^{N},{\mathbb {C}})$, as $n\rightarrow \infty$, then u is a weak solution of (36).

Lemma 21

If $(u_n)\subset H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}})$ is a sequence $(PS)_{c_{\lambda }}$ for $J_{A,V_{{\mathcal {P}}}}$ such that

$$\begin{aligned} u_n \rightharpoonup 0\quad \text {weakly in}\; H^1_{A,V_{{\mathcal {P}}}}({\mathbb {R}}^N,{\mathbb {C}}) \;\text { as}\; n\rightarrow \infty , \end{aligned}$$

with

$$\begin{aligned} c_{\lambda }<\frac{N+2-\alpha }{2(2N- \alpha )}S_{A}^{\frac{2N-\alpha }{N+2-\alpha }}, \end{aligned}$$

then there exist a sequence $(y_n)\in {\mathbb {R}}^N$ and constants $R,\theta >0$ such that

$$\begin{aligned}\limsup _{n\rightarrow \infty }\int \limits _{B_r(y_n)}|u_n|^2 \; \text {d}x\ge \theta ,\end{aligned}$$

where $B_r(y)$ denotes the ball in ${\mathbb {R}}^N$ of center at y and radius $r>0$.

The proof of Lemmas 19, 20 and 21 is similar to that of Corollary 9, Lemmas 10 and 12, respectively.

Lemma 22

Let $1<p<2^* -1$ and $u_{\varepsilon }$ as defined in (21). Then, there exists $\varepsilon$ such that

$$\begin{aligned} \sup _{t\ge 0}J_{A,V_{{\mathcal {P}}}}(tu_{\varepsilon })<\frac{N+2-\alpha }{2(2N-\alpha )}(S_{A})^{\frac{2N-\alpha }{N+2-\alpha }}, \end{aligned}$$

provided that either

(i):

$3<p<5$, $N=3$ and $\lambda >0;$

(ii):

$p>1$, $N\ge 4$ and $\lambda >0$;

(iii):

$1<p\le 3$, $N=3$ and $\lambda$ sufficiently large.

Proof

Consider, for cases (i) and (ii) the function $f:[0,+\infty )\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned}f(t)=J_{A,V_{{\mathcal {P}}}}(tu_{\varepsilon })=\frac{t^2}{2}\Vert u_{\varepsilon }\Vert ^2_{A,V_{{\mathcal {P}}}}-\frac{t^{2\cdot 2_{\alpha }^*}}{2\cdot 2_{\alpha }^*}D(u_{\varepsilon }) -\frac{\lambda t^{p+1}}{p+1}\Vert u_{\varepsilon }\Vert _{p+1}^{p+1}\end{aligned}$$

and proceed as in the proof of Case 1 and Lemma 14.

In the case of $1<p\le 3$, $N=3$ and $\lambda$ sufficiently large, consider $g_\lambda :[0,+\infty )\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned}g_{\lambda } (t)= J_{A,V_{{\mathcal {P}}}}(t u_{\varepsilon })= \frac{t^2}{2}\int \limits _{{\mathbb {R}}^N} \left[ |\nabla u_{\varepsilon }|^2+\left( |A(x)|^2+V_{{\mathcal {P}}}(x)\right) |u_{\varepsilon }|^2\right] \text {d}x-\frac{1}{2\cdot 2_{\alpha }^*}t^{2\cdot 2_{\alpha }^*} D(u_{\varepsilon })-\frac{\lambda t^{p+1}}{p+1}\Vert u_{\varepsilon }\Vert _{p+1}^{p+1}\end{aligned}$$

and proceed as in the proof of Case 2 and Lemma 14. $\square$

Similar to the proof of Theorem 13, we now state our result about the periodic problem (36).

Theorem 23

Under the hypotheses already stated on A and $\alpha$, suppose that $(V_1)$ is valid. Then, problem (36) has at least one ground-state solution if either

(i):

$3<p<5$, $N=3$ and $\lambda >0;$

(ii):

$p>1$, $N\ge 4$ and $\lambda >0$;

(iii):

$1<p\le 3$, $N=3$ and $\lambda$ sufficiently large.

4.2 Proof of Theorem 2

Some arguments of this proof were adapted from the proof of Theorem 1 that in turn were adapted from articles [2, 30].

Maintaining the notation already introduced, consider the functional $I_{A,V}:H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} I_{A,V}(u):=\frac{1}{2}{\Vert u\Vert ^2}_{A,V} -\frac{1}{2\cdot 2_{\alpha }^*}D(u)-\frac{\lambda }{p+1}\Vert u\Vert ^{p+1}_{p+1} \end{aligned}$$

for all $u\in H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})$.

We denote by ${\mathcal {N}}_{A,V}$ the Nehari Manifold related to $I_{A,V}$, that is,

$$\begin{aligned}{\mathcal {N}}_{A,V}&=\left\{ u\in H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})\setminus \{0\}\,:\,{\Vert u\Vert }^2_{A,V}=D(u)+\lambda \Vert u\Vert ^{p+1}_{p+1}\right\} , \end{aligned}$$

which is nonempty as a consequence of Theorem 23. As before, the functional $I_{A,V}$ satisfies the mountain pass geometry. Thus, there exists a $(PS)_{d_{\lambda }}$ sequence $(u_n)\subset H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})$, that is, a sequence satisfying

$$\begin{aligned}I'_{A,V}(u_n)\rightarrow 0\qquad \text {and}\qquad I_{A,V}(u_n)\rightarrow d_{\lambda },\end{aligned}$$

where $d_{\lambda }$ is the minimax level, also characterized by

$$\begin{aligned} d_{\lambda }=\inf _{u\;\in H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})\setminus \{0\}}\max _{t\ge 0}I_{A,V}(tu)=\inf _{{\mathcal {N}}_{A,V}} I_{A,V}(u)>0. \end{aligned}$$

As in Section 3, we have $I_{A,V}(u)<J_{A,V_{{\mathcal {P}}}}(u)$ for all $u\in H^1_{A,V}({\mathbb {R}}^N,{\mathbb {C}})$ as a consequence of ($V_2$).

Similar to the proof of Lemma 17 we have the following conclusion that shows an important inequality involving the levels $d_{\lambda }$ and $c_{\lambda }$, which completes the proof of Theorem 2.

Lemma 24

The levels $d_{\lambda }$ and $c_{\lambda }$ verify the inequality

$$\begin{aligned}d_{\lambda }<c_{\lambda } <\frac{N+2-\alpha }{2(N-\alpha )}(S_{A})^{\frac{2N-\alpha }{N+2-\alpha }}\end{aligned}$$

for all $\lambda >0$.

5 The case $f(u)=|u|^{2^{*}-2}u$

5.1 Proof of Theorem 3

As observed by Gao and Yang [25], the proof of Theorem 3 is analogous to the proof of Theorem 1. The principal distinction is that the $(PS)_{c_\lambda }$ condition holds true below the level $\frac{1}{N}S^{\frac{N}{2}}$. It follows from [37, Lemma 1.46] that

$$\begin{aligned} \displaystyle \int \limits _{{\mathbb {R}}^N} |\nabla u_{\varepsilon }|^2 \text {d}x=S^{\frac{N}{2}}+O(\varepsilon ^{N-2} ) \end{aligned}$$

and

$$\begin{aligned} \displaystyle \int \limits _{{\mathbb {R}}^N} | u_{\varepsilon }|^{2^{*}} \text {d}x=S^{\frac{N}{2}}+O(\varepsilon ^N ). \end{aligned}$$

So, we have

$$\begin{aligned} \displaystyle \sup _{t\ge 0}J_{A,V_{{\mathcal {P}}}}(t_{\varepsilon }u_{\varepsilon })&<\frac{1}{N} S^{\frac{N}{2}}+ O(\varepsilon ^{N-2})+ C_2\displaystyle \int \limits _{{\mathbb {R}}^N}|u_{\varepsilon }(x)|^2 \text {d}x-C_3 \varepsilon ^{2N-\alpha -(N-2)p}<\frac{1}{N}S^{\frac{N}{2}}, \end{aligned}$$

since

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\varepsilon ^{-(N-2)}\left( C_2\displaystyle \int \limits _{{\mathbb {R}}^N}|u_{\varepsilon }(x)|^2 \text {d}x-C_3\varepsilon ^{2N-\alpha -(N-2)p}\right) =-\infty . \end{aligned}$$

Observe that the last result is a consequence of

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\varepsilon ^{-(N-2)}\left( C_2\displaystyle \int \limits _{B_{\delta }}|u_{\varepsilon }(x)|^2 \text {d}x-C_3\varepsilon ^{2N-\alpha -(N-2)p}\right) =-\infty \end{aligned}$$

and

$$\begin{aligned}\ C_2\displaystyle \int \limits _{B_{2\delta }\setminus B_{\delta }}|u_{\varepsilon }(x)|^2 \text {d}x-C_3\varepsilon ^{2N-\alpha -(N-2)p}=O(\varepsilon ^{N-2}). \end{aligned}$$

The rest of the proof is omitted here. $\square$