Multiplicity of Normalized Solutions for Schrödinger Equations

Abstract

In this paper, we consider the following nonlinear Schrödinger equation with an $L^2$-constraint:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\lambda u+\mu |u|^{q-2}u+|u|^{p-2}u \ \ \ \textrm{in}~\mathbb {R}^{N}, \\ \int _{\mathbb {R}^{N}}|u|^{2}dx=a^2, \ \ u\in H^1(\mathbb {R}^{N}), \end{array}\right. }\end{aligned}$$

where $N\ge 3$, $a,\mu >0$, $2<q<2+\frac{4}{N}<p<2^*$, $2q+2N-pN<0$ and $\lambda \in \mathbb {R}$ arises as a Lagrange multiplier. We deal with the concave and convex cases of energy functional constraints on the $L^2$ sphere, and prove the existence of infinitely solutions with positive energy levels.

1 Introduction

In [21, 22], Soave considered the existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities

having prescribed mass

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

under different assumptions on $a>0, \mu \in \mathbb {R}$ and

$$\begin{aligned} 2<q \le 2+\frac{4}{N} \le p\le 2^*, \quad q \ne p, \end{aligned}$$

i.e. the two nonlinearities have different characters with respect to the $L^2$-critical exponent $\bar{p}:=2+\frac{4}{N}$. The cases $p>\bar{p}$ and $p<\bar{p}$ are called mass $L^{2}$-supercritical and mass $L^{2}$-subcritical, respectively, which comes from the Gagliardo-Nirenberg inequality (see [23]). We recall that, for every $N\ge 1$ and $p\in (2, 2^*)$, there exists a constant $C_{N,p}$ depending on N and on p such that

$$\begin{aligned} \Vert u\Vert _{p}^p\le C_{N,p}^p\Vert \nabla u\Vert _{2}^{p\delta _p}\Vert u\Vert _{2}^{p(1-\delta _p)}\quad \text {for all } u\in H^1(\mathbb {R}^N), \end{aligned}$$

(1.1)

where $\delta _p:=\frac{N( p - 2)}{2p}$ and we denote by $C_{N,p}$ the best constant in the Gagliardo-Nirenberg inequality.

Here and in what follows, $2^*$ denotes the critical exponent for the Sobolev embedding $H^1(\mathbb {R}^N ) \hookrightarrow L^p(\mathbb {R}^N )$ (that is, $2^* = 2N/(N -2)$ if $N \ge 3$ and $2^*=\infty$ if $N = 1, 2)$.

We look for solutions of problem ($\mathcal {P}$) having a prescribed $L^{2}$-norm, which are often referred to as normalized solutions. More precisely, for given $a>0$, we look for a couple of solution $(u_{a},\lambda _{a}) \in H^{1}(\mathbb {R}^{N}) \times \mathbb {R}$ to problem ($\mathcal {P}$) with

$$\begin{aligned} \int _{\mathbb {R}^N} |u_a|^{2}\textrm{d}x =a^2. \end{aligned}$$

The solution $u_a$ to the problem ($\mathcal {P}$) corresponds to a critical point of the following $C^1$ functional $\mathcal {J}: H_r^1(\mathbb {R}^N) \rightarrow \mathbb {R}$

$$\begin{aligned} \mathcal {J}(u)=\frac{1}{2}\Vert \nabla u\Vert _2^2-\frac{\mu }{p}\Vert u\Vert _p^p-\frac{1}{q}\Vert u\Vert _q^q \end{aligned}$$

restricted to the sphere in $L^2(\mathbb {R}^N)$:

$$\begin{aligned} \mathcal {S}(a) = \left\{ u \in H_r^1(\mathbb {R}^N): \Vert u\Vert _2^2 = a^2 \right\} . \end{aligned}$$

It is clear that for each critical point $u_{a}\in \mathcal {S}(a)$ of $\mathcal {J}|_{\mathcal {S}(a)}$ corresponds to a Lagrange multiplier $\lambda _{a} \in \mathbb {R}$ such that $(u_{a},\lambda _{a})$ solves problem ($\mathcal {P}$). Therefore, to obtain such a solution, it is necessary to find the critical point of $\mathcal {J}$ on the constraint $\mathcal {S}(a).$

In recent decades, the question of finding solutions of nonlinear Schrödinger equations with prescribed $L^2$-norm has received a special attention. This approach seems particularly meaningful from the physical point of view, since, in addition to being a conserved quantity for the time dependent equation, the mass has often a clear physical meaning; For instance, it represents the power supply in nonlinear optics, or the total number of atoms in Bose-Einstein condensation, two main fields of application of the nonlinear Schrödinger equations. For more related results on normalized solutions of nonlinear Schrödinger equations, we refer to [1, 5,6,7, 14,15,16,17, 20, 25, 26] and the references therein.

Notice that, for the case of $2< p< \bar{p} <q \le 2^*$, Soave [21, 22] applied the Gagliardo-Nirenberg inequality (1.1) to create the corresponding energy functional $h\in C^2(\mathbb {R}^+,\mathbb {R})$

$$\begin{aligned} h(t):=\frac{1}{2}t^2-\frac{\mu C_{N,q}^qa^{(1-\delta _q)q}}{q}t^{ q\delta _q}-\frac{ C_{N,p}^pa^{(1-\delta _p)p}}{p}t^{ p\delta _p} \end{aligned}$$

such that

$$\begin{aligned} \mathcal {J}(u)\ge \frac{1}{2}\Vert \nabla u\Vert _2^2-\frac{\mu C_{N,q}^qa^{(1-\delta _q)q}}{q}\Vert \nabla u\Vert _2^{ q\delta _q}-\frac{ C_{N,p}^pa^{(1-\delta _p)p}}{p}\Vert \nabla u\Vert _2^{ p\delta _p}=h(\Vert \nabla u\Vert _2). \end{aligned}$$

Recalling that $2<q<\bar{p}< p \le 2^*$, so that $q\delta _q < 2$ and $2<p\delta _p \le 2^*$. Under certain conditions of $a>0$ and $\mu >0$, function h has a concave-convex structure (see Fig. 1).

Fig. 1

The relationship between functional h and t.

Naturally, it is known that $\mathcal {J}$ has local minima and mountain path geometric structures on the constraint $\mathcal {S}(a).$ Therefore, Soave proved the existence of mountain pass solutions and local minimum solutions. After that, Alves, Ji and Miyagaki [3] studies problem ($\mathcal {P}$) with $p\in (2+\frac{4}{N},2^*)$, $q=2^*$ and $N\ge 3$, there exists $\mu ^*>0$ such that problem ($\mathcal {P}$) admits a couple $(u_a,\lambda _a)\in H^1(\mathbb {R}^N)\times \mathbb {R}^-$ of weak solutions, where $u_a$ is a radial, positive ground state solution of problem ($\mathcal {P}$) on $\mathcal {S}(a)$. In particular, replace $\mu |u|^{p-2} u+|u|^{q-2} u$ by f(u) such that f satisfies some critical growth conditions with $N=2$, then problem ($\mathcal {P}$) admits a couple $(u_a,\lambda _a)\in H^1(\mathbb {R}^2)\times \mathbb {R}^-$ of weak solutions. Subsequently, Alves, Ji and Miyagaki in [2] introduced a truncation function in $\mathcal {J}$. Precisely, they considered the truncated functional

$$\begin{aligned} {\mathcal {J}}_{\chi }(u)=\frac{1}{2}\Vert \nabla u\Vert _2^2-\frac{\mu }{q}\Vert u\Vert _q^q-\frac{\chi \left( \Vert \nabla u\Vert _2\right) }{p}\Vert u\Vert _p^p, \end{aligned}$$

where $\chi \in C^\infty _0(\mathbb {R}^{+}, [0,1])$ is nonincreasing and satisfies

$$\begin{aligned} \chi (t)= {\left\{ \begin{array}{ll} 1, \ \ \ \ &{} t\in [0, R_0], \\ 0, \ \ \ \ {} &{} t \in [R_1,\infty ). \end{array}\right. } \end{aligned}$$

Here $R_0$ and $R_1$ are given as in Fig. 1. Also by applying the Gagliardo-Nirenberg inequality (1.1), one shows that

$$\begin{aligned} \mathcal {J}_{\chi }(u) \ge h_1(\Vert \nabla u\Vert _2), \end{aligned}$$

where

$$\begin{aligned} {h}_1(t):=\frac{1}{2}t^2-\frac{\mu C_{N,q}^qa^{(1-\delta _q)q}}{q}t^{ q\delta _q}-\frac{ C_{N,p}^pa^{(1-\delta _p)p}}{p}\chi (t)t^{ p\delta _p}. \end{aligned}$$

Under certain conditions of $a>0$ and $\mu >0$, from Fig. 1 the image of ${h}_1$ is as Fig. 2, which implies that Alves, Ji and Miyagaki in [2] use a minimax theorem for a class of constrained even functionals that is proved in Jeanjean and Lu [16] to obtain the multiplicity of the solution of the energy functional ${\mathcal {J}}_\chi$ at the negative energy level. In fact, if $\mathcal {J}_\chi (u) \le 0$ then $\Vert \nabla u\Vert _2<R_0$, and $\mathcal {J}(v)=\mathcal {J}_\chi (v)$, for all v in a small neighborhood of u in $H^1\left( \mathbb {R}^N\right)$. Therefore, here the critical points of ${\mathcal {J}}_\chi$ are also are actually the critical points of ${\mathcal {J}}$.

Fig. 2

The relationship between functional $h_1$ and t.

In addition, this approach turns out to be useful also from the purely mathematical perspective, since it gives a better insight of the properties of the stationary solutions for ($\mathcal {P}$), such as stability or instability, can see [9, 10, 21, 22] and the references therein.

Moreover, we refer to [8], where Bartsch and Willem considered the model problem

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta u=\mu |u|^{q-2} u+\lambda |u|^{p-2} u, \\ u \in H_0^1(\Omega ), \end{array}\right. \end{aligned}$$

(1.2)

where $\Omega$ is a domain of $\mathbb {R}^N$ and $1<q<2<p<2^*$. The corresponding energy is defined on $H_0^1(\Omega )$ by

$$\begin{aligned} \varphi _{\lambda , \mu }(u):=\int _{\Omega }\left[ \frac{|\nabla u|^2}{2}-\frac{\mu |u|^q}{q}-\frac{\lambda |u|^p}{p}\right] d x. \end{aligned}$$

Then they showed that

• For every $\lambda >0, \mu \in \mathbb {R}$, problem (1.2) has a sequence of solutions $\left\{ u_k\right\}$ such that $\varphi _{\lambda , \mu }\left( u_k\right) \rightarrow \infty , k \rightarrow \infty$;

• For every $\mu >0, \lambda \in \mathbb {R}$, problem (1.2) has a sequence of solutions $\left\{ v_k\right\}$ such that $\varphi _{\lambda , \mu }\left( v_k\right) <0$ and $\varphi _{\lambda , \mu }\left( v_k\right) \rightarrow 0, k \rightarrow \infty$.

Inspired by above results, a natural guess that problem ($\mathcal {P}$) also possesses an unbounded sequence of solutions $\{(u_k,\lambda _k)\}\subset H^1(\mathbb {R}^N)\times \mathbb {R}^-$ with $\Vert u_k\Vert _2^2=a^2$ for each $k\in \mathbb {N}^+$, $\Vert \nabla u_k\Vert _2^2\rightarrow +\infty$ and $\mathcal {J}(u_k) \rightarrow +\infty$ as $k\rightarrow +\infty$. So, in this article we attempted to provide a positive answer. The main results of this paper are stated as:

Theorem 1.1

Assume that $2< q< \bar{p}<p < 2^*$, $2q+2N-pN<0$ and $N \ge 3$. For $a > 0$ and $\mu >0$ let us also suppose that

and

where $\beta _{max}$ defined in (2.1), then problem ($\mathcal {P}$) possesses an unbounded sequence of pairs of radial solutions $\{(u_k,\lambda _k)\}\subset H^1(\mathbb {R}^N)\times \mathbb {R}^-$ with $\Vert u_k\Vert _2^2=a^2$ for each $k\in \mathbb {N}^+$, $\Vert \nabla u_k\Vert _2^2\rightarrow +\infty$ and $\mathcal {J}(u_k) \rightarrow +\infty$ as $k\rightarrow +\infty$.

It is reasonable to assume (A.1) and (A.2) in the Theorem 1.1, because when we have $(p\delta _p-2)(1-\delta _q)q-q\delta _q(1-p)\delta _p>0$ at $2q+2N-pN<0$, then there is $a>0$ satisfying both assume (A.1) and (A.2).

Compared with [8], which works in a bounded region and has some compactness, while we work in the whole space and lack compactness, so additional restrictions are needed to ensure compactness. In addition, our method of proving is different, we are not a direct generalization of [8]. Compared with [2], we both adopted the truncation method, but the parts we truncated were different. On the one hand, the energy functional has an extra local term by truncating, which brings some difficulties to our subsequent estimation and compactility proof, while in [2], we can clearly see that when $\Vert \nabla u\Vert _2^2$ is bounded, $\mathcal {J}_\chi =\mathcal {J}$ is satisfied, that is, the local term has no effect on the proof. On the other hand, $\Vert \nabla u\Vert _2^2$ has at least k solutions at the negative energy level, and we have multiple solutions at the positive energy level.

In the proof of Theorem 1.1 we shall work on the space $H_r^1(\mathbb {R}^N )$, because it has a compact embedding. Moreover, by Palais’ principle of symmetric criticality, see [18], we know that the critical points of $\mathcal {J}$ in $H_r^1(\mathbb {R}^N )$ are in fact critical points in whole $H^1(\mathbb {R}^N )$. To prove the Theorem 1.1 we shall adapt for our case a truncation function found in Peral Alonso [19, Chapter 2, Theorem 2.4.6].

2 Preliminaries

We recall the functional h:

$$\begin{aligned} h(t):=\frac{1}{2}t^2-\frac{\mu C_{N,q}^qa^{(1-\delta _q)q}}{q}t^{ q\delta _q}-\frac{ C_{N,p}^pa^{(1-\delta _p)p}}{p}t^{ p\delta _p}. \end{aligned}$$

Since $a>0, \mu >0$ and $q\delta _q< 2 < p\delta _p$, we have that $h(0^+) = 0^-$ and $h(+\infty ) = -\infty$. The following proposition states the role of assumption (A.1).

Proposition 2.1

([21, See Lemma 5.1.]) Under assumption (A.1), the function h has a local strict minimum at negative level and a global strict maximum at positive level. Moreover, there exist $0< R_0< R_1 <\infty$, depending on $a>0$ and $\mu >0$, such that $h(R_0) = 0 = h(R_1)$ and $h(t) > 0$ iff $t \in (R_0, R_1)$, (see Fig. 1).

Under assumptions to (A.1), the function $\hat{h}$ has a global strict maximum at positive level, and there exist $0< R_1< R_2 <\infty$, depending on $a>0$, such that $\hat{h}(R_2) = 0$, where

$$\begin{aligned}\hat{h}(t)=\frac{1}{2}t^2-\frac{C_{N,p}^pa^{(1-\delta _p)p}}{p}t^{p\delta _p},\end{aligned}$$

and

$$\begin{aligned}R_2=\left( \frac{p}{2C_{N,p}^p}\right) ^{\frac{1}{p\delta _p-2}}a^{-\frac{(1-\delta _p)p}{p\delta _p-2}}.\end{aligned}$$

For $0< R_0< R_1 <\infty$, fix $\tau : \mathbb {R}^+ \rightarrow [0, 1]$ as being a $C^\infty$ function that satisfies

$$\begin{aligned}\tau (x)= {\left\{ \begin{array}{ll} 0~\text { if}&{}x\le R_0,\\ 1~\text { if}&{}x\ge R_1. \end{array}\right. }\end{aligned}$$

From proposition 2.1 we know that for the function h, we have $R_0$ and $R_1$ dependent of $a>0$ and $\mu >0$ such that $h_1(R_0) = 0 = h_1(R_1)$ and $h(t) > 0$ iff $t \in (R_0, R_1)$. For any fix $\mu >0$, we define the following functional H, denoted by

$$\begin{aligned}H(R,a)=\frac{1}{2}R^2-\frac{ \mu C_{N,q}^qa^{(1-\delta _q)q}}{q}R^{q\delta _ q}-\frac{C_{N,p}^pa^{(1-\delta _p)p}}{p}R^{p\delta _p}=h(R).\end{aligned}$$

For any $a_1,a_2>0$ that satisfies $a_1>a_2$, there is obviously

$$\begin{aligned}H(R_0(a_2),a_1)>H(R_0(a_2),a_2)=0=H(R_1(a_2),a_2)<H(R_1(a_2),a_1).\end{aligned}$$

According to the structure of functional h, we can obtain

$$\begin{aligned}R_0(a_2)>R_0(a_1)~~\text {and}~~R_1(a_2)<R_1(a_1),\end{aligned}$$

therefore, $a\mapsto \mathcal {R}(a):=R_1(a)-R_0(a)$ is non-increasing and under the assumption (A.1) $\mathcal {R}(a)$ has a lower bound $\alpha >0$. Now, under assumption (A.1), for any $a>0$ we can take $\tau$ such that $\tau '$ has a uniform upper bound, and we remember that the uniform upper bound is $\beta _{1}$, where we have $\tau '(x)\in [0,\beta _{1})$ when $x\in [0,\infty )$ (Rule out that if $a>0$ is small enough it may not be possible to find $\tau$ such that $\tau '$ has no uniform upper bound).

By the same token, we have a similar conclusion for any fixed $a>0$, for any $\mu >0$ we can take $\tau$ such that $\tau '$ has a uniform upper bound $\beta _2$. Therefore, for any $a>0$ and $\mu >0$ we can take $\tau$ such that $\tau '$ has a uniform upper bound under the assumption (A.1), which we remember

$$\begin{aligned} \beta _{max}:=\max \{\beta _1,\beta _2\}. \end{aligned}$$

(2.1)

Thus we have $\tau '(x) \in [0, \beta _ {max})$ when $x \in [0, \infty )$.

In the sequel, let us consider the truncated functional

$$\begin{aligned}\mathcal {J}_T(u)=\frac{1}{2}\Vert \nabla u\Vert _2^2-\frac{\mu \tau (\Vert \nabla u\Vert _2)}{p}\Vert u\Vert _q^q-\frac{1}{p}\Vert u\Vert _p^p.\end{aligned}$$

Thus

$$\begin{aligned}\mathcal {J}_T(u)\ge h_2(\Vert \nabla u\Vert _2),\end{aligned}$$

where

$$\begin{aligned}h_2(t)=\frac{1}{2}t^2-\frac{ \mu C_{N,q}^qa^{(1-\delta _q)q}}{q}t^{q\delta _ q}\tau (t)-\frac{C_{N,p}^pa^{(1-\delta _p)p}}{p}t^{p\delta _p},\quad \text {(see Fig. }3).\end{aligned}$$

Fig. 3

The relationship between functional $h_2$ and t.

The truncated functional $\mathcal {J}_T$ and the relationship between $\mathcal {J}_T$ and $\mathcal {J}$ are noteworthy. By reference [2, See Lemma 3.1] we get the following lemma.

Lemma 2.1

Assume that $N \ge 3, 2< q< \bar{p}<p < 2^*$ and (A.1) holds, then the functional $\mathcal {J}_T$ has some important properties:

(i):

$\mathcal {J}_T\in C^1(H_r^1(\mathbb {R}^N),\mathbb {R})$.

(ii):

If $\mathcal {J}_T\le 0$ then $\Vert \nabla u\Vert _2^2\ge R_1$, and $\mathcal {J}(v)=\mathcal {J}_T(v)$, for all v in a small neighborhood of u in $H_r^1(\mathbb {R}^N)$.

In order to recover some compacity, we will work in $E = H_r^1(\mathbb {R}^N )$, provided with the standard scalar product and norm: $\Vert u\Vert _H^2 =\Vert \nabla u\Vert _2^2 + \Vert u\Vert _2^2$. Here and in the sequel we write $\Vert u\Vert _p^p$ to denote the $L^p$-norm. For convenience, $C_1,C_2,\cdots$ denote various positive constants.

3 Proof of Theorem 1.1

To prove our conclusion, we use the proof technique in [5], but here our nonlinear term does not satisfy the conditions in [5]. The main theorem’s proof will follow from several lemmas. We fix a strictly increasing sequence of finite-dimensional linear subspaces $V_n\subset E$ such that $\bigcup _n V_n$ is dense in E.

Lemma 3.1

( [5, See Lemma 2.1.]) For $2< r < 2^*$ there holds:

$$\begin{aligned}\mu _n(r)=\inf \limits _{u\in V_{n-1}^\bot }\dfrac{\int _{\mathbb {R}^N}(|\nabla u|^2+|u|^2)dx}{(\int _{\mathbb {R}^N}|u|^rdx)^{2/r}}=\inf \limits _{u\in V_{n-1}^\bot }\dfrac{\Vert u\Vert _H^2}{\Vert u\Vert _r^2}\rightarrow \infty ~\text {as}~n\rightarrow \infty .\end{aligned}$$

Here we give the following definition

$$\begin{aligned} \int _{\mathbb {R}^N}F(u)dx=\frac{\mu \tau (\Vert \nabla u\Vert _2)}{q}\Vert u\Vert _q^q+\frac{1}{p}\Vert u\Vert _p^p. \end{aligned}$$

(3.1)

We introduce now the constant

$$\begin{aligned}K=\max \limits _{u\in H_r^1(\mathbb {R}^N)}\dfrac{\int _{\mathbb {R}^N}F(u)dx}{\Vert u\Vert _p^p+\Vert u\Vert _q^q},\end{aligned}$$

which is well defined when combined with (3.1). For $n \in \mathbb {N}$ we define

$$\begin{aligned}\rho _n=\dfrac{M_n^{p/(p-2)}}{L^{2/(p-2)}},\end{aligned}$$

where

$$\begin{aligned}M_n=[\mu _n(q)^{-q/2}+\mu _n(p)^{-p/2}]^{-2/p}\quad \text {and}\quad L=3K\max \limits _{\theta >0}\dfrac{(a+\theta ^2)^{p/2}}{a+\theta ^{p}}.\end{aligned}$$

By Lemma 3.1 we have $\rho _n\rightarrow \infty$ as $n\rightarrow \infty$. We also define

$$\begin{aligned}B_n:=\left\{ u\in V_{n-1}^\bot \cap \mathcal {S}(a): \Vert \nabla u\Vert _2^2=\rho _n\right\} ,\end{aligned}$$

here $V_{n-1}^\bot$ is the orthogonal complement of $V_{n-1}$. Then we have:

Lemma 3.2

$b_n=\inf _{u\in B_n}\mathcal {J}_T(u)\rightarrow \infty ~\text {as}~n\rightarrow \infty .$

Proof

For any $a>0$ and $u\in B_n$, because of $\rho _n\rightarrow \infty$ as $n\rightarrow \infty$, we have $\Vert \nabla u\Vert _2^2+a^2>1$ when n is large enough. Since $2< q<\bar{p}<p < 2^*$ we deduce using the preceding lemma with $r=p$ and $r=q$,

$$\begin{aligned} \mathcal {J}_T(u)= & {} \frac{1}{2}\Vert \nabla u\Vert _2^2-\frac{\mu \tau (\Vert \nabla u\Vert _2)}{q}\Vert u\Vert _q^q-\frac{1}{p}\Vert u\Vert _p^p\\\ge & {} \frac{1}{2}\Vert \nabla u\Vert _2^2-K\Vert u\Vert _q^q-K\Vert u\Vert _p^p\\\ge & {} \frac{1}{2}\Vert \nabla u\Vert _2^2-\frac{K}{\mu _n(q)^{q/2}}\left( \Vert \nabla u\Vert _2^2+a^2\right) ^{q/2} -\frac{K}{\mu _n(p)^{p/2}}\left( \Vert \nabla u\Vert _2^2+a^2\right) ^{p/2}\\\ge & {} \frac{1}{2}\Vert \nabla u\Vert _2^2-K\left[ \mu _n(q)^{-q/2} +\mu _n(p)^{-p/2}\right] \left( \Vert \nabla u\Vert _2^2+a^2\right) ^{p/2}\\\ge & {} \frac{1}{2}\Vert \nabla u\Vert _2^2-\frac{K}{M_n^{p/2}}\left( \Vert \nabla u\Vert _2^2+a^2\right) ^{p/2}\\\ge & {} \frac{1}{2}\Vert \nabla u\Vert _2^2-\frac{L}{3M_n^{p/2}}\left( \Vert \nabla u\Vert _2^p+a^2\right) \\\ge & {} \frac{1}{2}\rho _n-\frac{L}{3M_n^{p/2}}\rho ^{p/2}_n\\= & {} \left( \frac{1}{2}-\frac{1}{3}\right) \rho _n\rightarrow \infty . \end{aligned}$$

The proof of the lemma is complete. $\square$

Lemma 3.3

Assume that $N\ge 3$ and $2< q<\bar{p}<p< 2^*$. Then there exists $0< \rho _0 < R_0^2$ such that

$$\begin{aligned} 0<\sup \limits _{u\in M_1}\mathcal {J}_T(u)<b_0:=\inf \limits _{u\in M_2}\mathcal {J}_T(u), \end{aligned}$$

(3.2)

where

$$\begin{aligned} M_1:=\left\{ u \in \mathcal {S}(a),\Vert \nabla u\Vert _2^2 \le \rho _0/2\right\} ,~~ M_2:=\left\{ u \in \mathcal {S}(a),\Vert \nabla u\Vert _2^2=\rho _0\right\} . \end{aligned}$$

Proof

Now, using equation (3.1) and Gagliardo-Nirenberg inequality (1.1) and taking into account that $\Vert u\Vert _2^2=a^2$

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^N}F(u)dx&=\frac{\mu \tau (\Vert \nabla u\Vert _2)}{q}\Vert u\Vert _q^q+\frac{1}{p}\Vert u\Vert _p^p\\&\le \frac{\mu C_{N,q}^qa^{(1-\delta _q)q}}{q}\Vert \nabla u\Vert _2^{q\delta _ q}\tau (\Vert \nabla u\Vert _2)+\frac{C_{N,p}^pa^{(1-\delta _p)p}}{p}\Vert \nabla u\Vert _2^{p\delta _p}. \end{aligned} \end{aligned}$$

Then we have for $\Vert \nabla u\Vert _2<R_0$ small enough,

$$\begin{aligned} \int _{\mathbb {R}^N}F(u)dx\le \frac{C_{N,p}^pa^{(1-\delta _p)p}}{p}\Vert \nabla u\Vert _2^{p\delta _p}. \end{aligned}$$

(3.3)

Next, let $0< \rho < R_0$ be arbitrary but fixed and suppose $u, v \in \mathcal {S}(a)$ are such that $\Vert \nabla u\Vert _2^2\le \rho /2$ and $\Vert \nabla v\Vert _2^2=\rho$. Then, for $\rho >0$ small enough

$$\begin{aligned} \begin{aligned} \mathcal {J}_T(v)-\mathcal {J}_T(u)&=\frac{1}{2}\Vert \nabla v\Vert _2^2-\frac{1}{2}\Vert \nabla u\Vert _2^2-\int _{\mathbb {R}^N}F(v)dx+\int _{\mathbb {R}^N}F(u)dx \\&\ge \frac{\rho }{4}-\int _{\mathbb {R}^N} F(v) dx \\&\ge \frac{\rho }{4}-C \rho ^{p\delta _p}\\&\ge \frac{\rho }{8}, \end{aligned} \end{aligned}$$

using (3.3) and $p\delta _p=\frac{N(q-2)}{2} > 2$. The proof of the lemma is complete. $\square$

In order to set up a min-max scheme, let

$$\begin{aligned}\varphi : \mathbb {R} \times E\rightarrow E,~ \varphi (s, u) = s *u,\end{aligned}$$

be the action of the group $\mathbb {R}$ on E defined by

$$\begin{aligned}(s*u)(x) = e^{sN/2}u(e^sx)~ \text { for } s \in \mathbb {R}, u \in E, x \in \mathbb {R}^N.\end{aligned}$$

Observe that $s*u\in \mathcal {S}(a)$ if $u\in \mathcal {S}(a)$, and that for $u \in \mathcal {S}(a)$

$$\begin{aligned} \Vert \nabla (s*u)\Vert _2^2\rightarrow 0,~ \mathcal {J}_T(s*u)\rightarrow 0\quad \text {as }~s\rightarrow -\infty . \end{aligned}$$

(3.4)

Moreover, $\int _{\mathbb {R}^N}F(u)dx \ge \frac{1}{p}\Vert u\Vert _p^p$ for all $u\in H^1(\mathbb {R}^N)$, and therefore

$$\begin{aligned} \begin{aligned} \mathcal {J}_T(s*u)&=\frac{1}{2}\Vert \nabla (s*u)\Vert _2^2-\int _{\mathbb {R}^N}F(s*u)dx\\&\le \frac{1}{2}\Vert \nabla (s*u)\Vert _2^2-\frac{1}{p}\Vert s*u\Vert _p^p\\&=\frac{e^{2s}}{2}\Vert \nabla u\Vert _2^2-\frac{e^{-Ns}e^{(Nsp)/2}}{p}\Vert u\Vert _p^p\rightarrow -\infty ~~\text {as}~~s\rightarrow \infty , \end{aligned} \end{aligned}$$

because $-Ns + (Nsp) /2 > 2s$. As a consequence we obtain for $u \in \mathcal {S}(a)$ that

$$\begin{aligned} \Vert \nabla (s*u)\Vert _2^2\rightarrow \infty ,~\mathcal {J}_T(s *u)\rightarrow -\infty ~~\text {as}~~s\rightarrow \infty . \end{aligned}$$

(3.5)

Due to (3.4) and (3.5), there exists $s_n > 0$ such that

$$\begin{aligned}\tilde{\gamma }_n: [ 0, 1] \times (\mathcal {S}(a) \cap V_n) \rightarrow \mathcal {S}(a),~ \tilde{\gamma }_n(t, u) = ( 2s_nt - s_n) *u,\end{aligned}$$

satisfies (with $\rho _0, b_0$ from Lemma 3.3, $b_n$ from Lemma 3.2):

$$\begin{aligned} \Vert \nabla \tilde{\gamma }_n(0,u)\Vert _2^2< \rho _0 < \rho _n, ~\Vert \nabla \tilde{\gamma }_n(1,u)\Vert _2^2> \rho _n, \end{aligned}$$

and

$$\begin{aligned} 0< \mathcal {J}_T(\tilde{\gamma }_n(0,u))< \max \{b_0, b_n\},~ \mathcal {J}_T(\tilde{\gamma }_n(1,u))< b_n, \end{aligned}$$

(3.6)

uniformly for $u\in \mathcal {S}(a) \cap V_n$. Now we define

$$\begin{aligned} \Gamma _n:=\left\{ \gamma : [ 0, 1] \times (\mathcal {S}(a) \cap V_n) \rightarrow \mathcal {S}(a) \left| \begin{aligned}&\gamma ~\text {is continuous, odd in } u,\\&\gamma (0, u) = \tilde{\gamma }_n(0, u),~\gamma (1, u) = \tilde{\gamma }_n(1, u) \end{aligned} \right. \right\} . \end{aligned}$$

Clearly we have $\tilde{\gamma }_n \in \Gamma _n$. Here we define the mountain pass value

$$\begin{aligned} c_n = \inf \limits _{\gamma \in \Gamma _n} \max \limits _{\mathop {t\in [0,1]}\limits _{u\in \mathcal {S}(a) \cap V_n} }\mathcal {J}_T(\gamma (t, u)). \end{aligned}$$

For the sake of subsequent lemmas, in the following we recall some properties of the cohomological index for spaces with an action of the group $G = \{-1, 1\}$. This index goes back to [11] and has been used in a variational setting in [12]. It associates to a G-space X an element $i(X) \in \mathbb {N}_0 \cup \{\infty \}$. We only need the following properties.

$(I_1)$:

If G acts on $\mathbb {S}^{n-1}$ via multiplication, then $i(\mathbb {S}^{n-1}) = n$.

$(I_2)$:

If there exists an equivariant map $X \rightarrow Y,$ then $i(X) \le i(Y )$.

$(I_3)$:

Let $X = X_0 \cup X_1$ be metrisable and $X_0, X_1 \subset X$ be closed G-invariant subspaces. Let Y be a G-space, and consider a continuous map $\phi :[0, 1]\times Y \rightarrow X$ such that each $\phi _t = \phi (t, \cdot ): Y \rightarrow X$ is equivariant. If $\phi _0(Y ) \subset X_0$ and $\phi _1(Y ) \subset X_1$, then

$$\begin{aligned}i(\textrm{Im}(\phi ) \cap X_0 \cap X_1) \ge i(Y ).\end{aligned}$$

Properties $(I_1)$ and $(I_2)$ are standard and hold also for the Krasnoselskii genus. Property $(I_3)$ has been proven in [4, Corollary 4.11, Remark 4.12].

We now need the following linking property, and it is proved by the above properties.

Lemma 3.4

For every $\gamma \in \Gamma _n$, there exists $(t, u) \in [0,1] \times (\mathcal {S}(a) \cap V_n)$ such that $\gamma (t, u) \in B_n$.

Proof

Let $\mathcal {T}_{n-1}: E \rightarrow V_{n-1}$ be the orthogonal projection, and set

$$\begin{aligned}h_n: \mathcal {S}(a) \rightarrow V_{n-1} \times \mathbb {R}^+,~~ u \mapsto (\mathcal {T}_{n-1}u, \Vert \nabla u\Vert _2^2).\end{aligned}$$

Then clearly $B_n = h_n^{-1}(0, \rho _n)$. We fix $\gamma \in \Gamma _n$ and consider the map

$$\begin{aligned}\phi =h_n\circ \gamma :[0,1]\times (\mathcal {S}(a) \cap V_n)\rightarrow V_{n-1}\times \mathbb {R}^+=:X.\end{aligned}$$

Since

$$\begin{aligned}\phi _0(\mathcal {S}(a) \cap V_n)\subset V_{n-1}\times (0,\rho _n]=:X_0\end{aligned}$$

and

$$\begin{aligned}\phi _1(\mathcal {S}(a) \cap V_n)\subset V_{n-1}\times (\rho _n,\infty ]=:X_1,\end{aligned}$$

it follows from $(I_1)$ to $(I_3)$ that

$$\begin{aligned}i(\textrm{Im}(\phi ) \cap X_0 \cap X_1) \ge i(\mathcal {S}(a) \cap V_n )=\textrm{dim} V_n.\end{aligned}$$

If there would not exist $(t, u) \in [0, 1] \times (\mathcal {S}(a) \cap V_n)$ with $\gamma (t, u) \in B_n$, then

$$\begin{aligned}\textrm{Im}(\phi ) \cap X_0 \cap X_1 \subset (V_{n-1}\setminus {0}) \times \{\rho _n\}.\end{aligned}$$

Now $(I_1)$ and $(I_2)$ imply that

$$\begin{aligned}i(\textrm{Im}(\phi ) \cap X_0 \cap X_1) \le i((V_{n-1}\setminus {0})\times \{\rho _n\}) = \textrm{dim} V_{n-1},\end{aligned}$$

contradicting $\textrm{dim} V_{n-1} < \textrm{dim} V_{n}.$ $\square$

It follows from Lemma 3.3 that

$$\begin{aligned} c_n = \inf \limits _{\gamma \in \Gamma _n} \max \limits _{\mathop {t\in [0,1]}\limits _{u\in \mathcal {S}(a) \cap V_n} }\mathcal {J}_T(\gamma (t, u)) \ge b_n = \inf \limits _{u\in B_n}\mathcal {J}_T(u)\rightarrow \infty . \end{aligned}$$

(3.7)

Clearly by (3.2) and (3.6) there also holds

$$\begin{aligned} c_n\ge b_0>0. \end{aligned}$$

(3.8)

We recall the stretched functional from [15], see also [13]:

$$\begin{aligned}\bar{\mathcal {J}}_T: \mathbb {R}\times E \rightarrow \mathbb {R},~~ (s, u) \mapsto \mathcal {J}_T(s *u).\end{aligned}$$

Now we define

$$\begin{aligned} \bar{\Gamma }_n:=\left\{ \bar{\gamma }:[ 0, 1] \times (\mathcal {S}(a) \cap V_n) \rightarrow \mathbb {R}\times \mathcal {S}(a) \left| \begin{aligned}&\bar{\gamma }~\text {is continuous, odd in } u,\\&\varphi \circ \bar{\gamma }\in \Gamma _n \end{aligned} \right. \right\} , \end{aligned}$$

where $\varphi (s,u)=s*u$ and

$$\begin{aligned}\bar{c}_n = \inf \limits _{\bar{\gamma }\in \bar{\Gamma }_n} \max \limits _{\mathop {t\in [0,1]}\limits _{u\in \mathcal {S}(a) \cap V_n} }\bar{\mathcal {J}}_T(\bar{\gamma }(t, u)).\end{aligned}$$

Reference [5, Lemma 2.5], we also have conclusions $c_n=\bar{c}_n$ for $c_n$ and $\bar{c}_n$.

Next, we will show that $c_n$ is a critical value of $\mathcal {J}_T$, which is an important part of the proof of Theorem 1.1. We fix n from now on.

Lemma 3.5

There exists a Palais-Smale sequence $\{u_k^n\}$ for $\mathcal {J}_T$ at the level $c_n$ satisfying $P(u_k^n)\rightarrow 0$ as $k\rightarrow \infty$, where

$$\begin{aligned} P(u)=\Vert \nabla u\Vert _2^2-\delta _q\mu \tau (\Vert \nabla u\Vert _2)\Vert u\Vert _q^q-\frac{\mu \tau '(\Vert \nabla u\Vert _2)}{q}\Vert \nabla u\Vert _2\Vert u\Vert _q^q-\delta _p\Vert u\Vert _p^p. \nonumber \\ \end{aligned}$$

(3.9)

Proof

For $\gamma \in \Gamma _n$ there holds by (3.6), (3.7), (3.8), and the definition of $\Gamma _n$:

$$\begin{aligned} \begin{aligned} c_n&\ge \max \left\{ b_0, b_n\right\} >\max \left\{ \max _{u \in \mathcal {S}(a) \cap V_n} \mathcal {J}_T\left( \tilde{\gamma }_n(0, u)\right) , \max _{u \in \mathcal {S}(a) \cap V_n} \mathcal {J}_T\left( \tilde{\gamma }_n(1, u)\right) \right\} \\&=\max \left\{ \max _{u \in \mathcal {S}(a) \cap V_n} \mathcal {J}_T(\gamma (0, u)), \max _{u \in \mathcal {S}(a) \cap V_n} \mathcal {J}_T(\gamma (1, u))\right\} . \end{aligned} \end{aligned}$$

Using the fact $c_n=\bar{c}_n$ we obtain a sequence $\left\{ \gamma _k^n\right\}$ in $\Gamma _n$ such that

$$\begin{aligned} \max _{[0,1] \times \left( \mathcal {S}(a) \cap V_n\right) } \bar{\mathcal {J}}_T\left( 0, \gamma _k^n\right) \rightarrow c_n. \end{aligned}$$

Now Ekeland’s variational principle implies the existence of a Palais-Smale sequence $\left\{ (s_k^n, u_k^n)\right\}$ for $\left. \bar{\mathcal {J}}_T\right| _{\mathbb {R} \times \mathcal {S}(a)}$ at the level $c_n$ such that $s_k^n \rightarrow 0$. From $\bar{\mathcal {J}}_T(s, u)=$ $\bar{\mathcal {J}}_T(0, s * u)$ and for every $\psi \in H^1(\mathbb {R}^N)$ we deduce

$$\begin{aligned} \left( \partial _s \bar{\mathcal {J}}_T\right) (s, u)=\left( \partial _s \bar{\mathcal {J}}_T\right) (0, s * u) \quad \text{ and } \quad \left( \partial _u \bar{\mathcal {J}}_T\right) (s, u)[\psi ]=\left( \partial _u \bar{\mathcal {J}}_T\right) (0, s * u)[s * \psi ] \end{aligned}$$

so that $\left\{ \left( 0, s_k^n * u_k^n\right) \right\}$ is also a Palais-Smale sequence for $\left. \bar{\mathcal {J}}_T\right| _{\mathbb {R} \times S}$ at the level $c_n$. Thus we may assume that $s_k^n=0$. This implies, firstly, that $\left\{ u_k^n\right\}$ is a PalaisSmale sequence for $\mathcal {J}_T$ at the level $c_n$, and secondly, using $\partial _s \bar{\mathcal {J}}_T\left( 0, u_k^n\right) \rightarrow 0$ as $k\rightarrow \infty$, that is $P(u_k^n)\rightarrow 0$ as $k\rightarrow \infty$. $\square$

Lemma 3.6

Assume that $N \ge 3$ and $2< q< \bar{p}<p < 2^*$. (A.2) is satisfied for any $a>0$ and $\mu >0$, if the sequence $\left\{ u_k\right\}$ in $\mathcal {S}(a)$ satisfies $\mathcal {J}_T^{\prime }\left( u_k\right) \rightarrow 0, \mathcal {J}_T\left( u_k\right) \rightarrow c>0$, and $P(u_k)\rightarrow 0$ as $k\rightarrow \infty$, then it is bounded in E and has a convergent subsequence.

Proof

Claim 1: The sequence $\left\{ u_k\right\}$ is bounded in E.

Suppose $\left\{ u_k\right\}$ is unbounded, that is, $\Vert \nabla u_k\Vert _2^2\rightarrow \infty$ as $k\rightarrow \infty$. As $P(u_k) \rightarrow 0$ as $k\rightarrow \infty$, we observe that

$$\begin{aligned}\Vert u_k\Vert _p^p=\frac{1}{\delta _p}\Vert \nabla u_k\Vert _2^2-\frac{\mu \delta _q\tau (\Vert \nabla u_k\Vert _2)}{\delta _p}\Vert u_k\Vert _q^q-\frac{\mu \tau '(\Vert \nabla u_k\Vert _2)}{q\delta _p}\Vert \nabla u_k\Vert _2\Vert u_k\Vert _q^q+o(1).\end{aligned}$$

Whence

$$\begin{aligned} \begin{aligned} \mathcal {J}_T(u_k)=&\left( \frac{1}{2}-\frac{1}{p\delta _p}\right) \Vert \nabla u_k\Vert _2^2-\frac{\mu }{q}\left( 1-\frac{q\delta _q}{p\delta _p}\right) \tau (\Vert \nabla u_k\Vert _2)\Vert u_k\Vert _q^q\\&+\frac{\mu \tau '(\Vert \nabla u_k\Vert _2)}{qp\delta _p}\Vert \nabla u_k\Vert _2\Vert u_k\Vert _q^q+o(1), \end{aligned} \end{aligned}$$

by the Gagliardo-Nirenberg inequality (1.1) and $\tau '(x)\in [0,\beta _{max})$ we have that

$$\begin{aligned} \begin{aligned} c+1&\ge \mathcal {J}_T(u_k)\ge \left( \frac{1}{2}-\frac{1}{p\delta _p}\right) \Vert \nabla u_k\Vert _2^2-\frac{\mu }{q}\left( 1-\frac{q\delta _q}{p\delta _p}\right) \tau (\Vert \nabla u_k\Vert _2)C_{N,q}^qa^{(1-\delta _q)q}\Vert \nabla u_k\Vert _2^{ q\delta _q}, \end{aligned} \end{aligned}$$

this implies that

$$\begin{aligned}\Vert \nabla u_k\Vert _2^2\le C a^{(1-\delta _q)q}\tau (\Vert \nabla u_k\Vert _2^2)\Vert \nabla u_k\Vert _2^{ q\delta _q}+C,\end{aligned}$$

since $q\delta _q < 2,$ the boundedness of $\{u_k\}$ follows also in this case.

As $\left\{ u_k\right\}$ is bounded in $H_r^1(\mathbb {R}^N)$, and $H_r^1(\mathbb {R}^N)\hookrightarrow L^l(\mathbb {R}^N)$ compactly for $l\in (2, 2^*)$, there exists $u\in H_r^1(\mathbb {R}^N)$ such that up to a subsequence

$$\begin{aligned}u_{k}\rightharpoonup u \text { in } H _r^{1}(\mathbb {R}^{N}),~ u_k\rightarrow u \text { in } L^l(\mathbb {R}^N) \text { and } u_k \rightarrow u ~a.e.\text { in }\mathbb {R}^N.\end{aligned}$$

Claim 2: The weak limit u is nontrivial, that is, $u\not \equiv 0.$

Since $\mathcal {J}_T(u_k) \rightarrow c \ne 0$, using the fact that $P(u_k) \rightarrow 0$ and $\tau '(x)\in [0,\beta _{max})\text { for all }x\in \mathbb {R}$, we had $u_k \rightarrow 0$ we would find by strong $L^p(\mathbb {R}^N)$ and $L^q(\mathbb {R}^N)$ convergence that

$$\begin{aligned} \begin{aligned} \mathcal {J}_T(u_k)&=\frac{\mu }{q}\left( \frac{q\delta _q}{2}-1\right) \tau (\Vert \nabla u_k\Vert _2)\Vert u_k\Vert _q^q+\frac{\mu \tau '(\Vert \nabla u_k\Vert _2)}{2q}\Vert \nabla u_k\Vert _2\Vert u_k\Vert _q^q\\&-\frac{1}{p}\left( \frac{p\delta _q}{2}-1\right) \Vert u_k\Vert _p^p+o(1)\rightarrow 0, \end{aligned} \end{aligned}$$

that is a contradiction.

Claim 3: $\lambda _k \rightarrow \lambda < 0$.

By Willem [24, Proposition 5.12], there exists $\{\lambda _k\}\subset \mathbb {R}$ such that

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^N}\nabla u_k\cdot \nabla \psi dx-\mu \tau (\Vert \nabla u_k\Vert _2)\int _{\mathbb {R}^{N}}|u_k|^{q-2}u_k\psi dx-\int _{\mathbb {R}^{N}}|u_k|^{p-2}u_k\psi dx \\&-\Vert u_k\Vert _p^p\frac{\mu \tau '(\Vert \nabla u_k\Vert _2)}{p}\left( \int _{\mathbb {R}^{N}}\nabla u_k\cdot \nabla \psi dx\right) ^{\frac{1}{2}}=\int _{\mathbb {R}^N}\lambda _k u_k\psi dx+o(1)\Vert \psi \Vert _H, \end{aligned} \end{aligned}$$

(3.10)

for every $\psi \in H^1(\mathbb {R}^N)$, where $o(1)\rightarrow 0$ as $n \rightarrow \infty$. The choice $\psi = u_k$ provides

$$\begin{aligned} \Vert \nabla u_k\Vert _2^2-\mu \tau (\Vert \nabla u_k\Vert _2)\Vert u_k\Vert _q^q-\frac{\mu \tau '(\Vert \nabla u_k\Vert _2)}{q}\Vert \nabla u_k\Vert _2\Vert u_k\Vert _q^q-\Vert u_k\Vert _p^p=\lambda _ka^2+o(1). \end{aligned}$$

Recalling that $P(u_k) \rightarrow 0$, we have

$$\begin{aligned} \lambda _ka^2=\mu (\delta _q-1)\tau (\Vert \nabla u_k\Vert _2)\Vert u_k\Vert _q^q+(\delta _p-1)\Vert u_k\Vert _p^p+o(1), \end{aligned}$$

(3.11)

since $0< \delta _q, \delta _p < 1$, we deduce that $\left\{ \lambda _k\right\}$ is bounded and $\lambda _k \le 0$. We now claim that

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\Vert \nabla u_k\Vert _2=A>0. \end{aligned}$$

If not, from Gagliardo-Nirenberg inequality (1.1) we obtain

$$\begin{aligned}\lim \limits _{k\rightarrow \infty }\int _{\mathbb {R}^{N}}|u_k|^ldx\rightarrow 0 ~\text { for }~l\in (2,2^*),\end{aligned}$$

then

$$\begin{aligned}0\ne c=\lim \limits _{n\rightarrow \infty }\mathcal {J}_T(u_k)=\lim \limits _{n\rightarrow \infty }\left[ \frac{1}{2}\Vert \nabla u_k\Vert _2^2-\frac{\mu \tau (\Vert \nabla u_k\Vert _2)}{q}\Vert u_k\Vert _q^q-\frac{1}{p}\Vert u_k\Vert _p^p\right] =0.\end{aligned}$$

Next, we proved that up to a subsequence $\lambda _k \rightarrow \lambda < 0$. Using the strong $L^p(\mathbb {R}^N)$ and $L^q(\mathbb {R}^N)$ convergence of $\{u_k\}$, by (3.11) we have that

$$\begin{aligned} \lambda a^2=(\delta _q-1)\tau (A)\Vert u\Vert _q^q+(\delta _p-1)\Vert u\Vert _p^p, \end{aligned}$$

since $0< \delta _q, \delta _p < 1$ and $\tau (A)\ge 0$, we must have $\lambda < 0$.

Claim 4: $u_k\rightarrow u$ in $H_r^1(\mathbb {R}^N)$.

Up to a subsequence, let $\lim _{n\rightarrow \infty }\Vert \nabla u_k\Vert _{2}^{2} = A^2>0$. Then, u satisfies

$$\begin{aligned} \Vert \nabla u\Vert _2^2-\mu \tau (A)\Vert u\Vert _q^q-\frac{\mu \tau '(A)}{q}\Vert \nabla u\Vert _2\Vert u\Vert _q^q-\Vert u\Vert _p^p=\lambda \Vert u\Vert _2^2. \end{aligned}$$

(3.12)

By (3.10) and (3.12), we obtain

$$\begin{aligned} \begin{aligned}&\Vert \nabla u\Vert _2^2-\mu \tau (A)\Vert u\Vert _q^q-\frac{\mu \tau '(A)}{p}\Vert \nabla u\Vert _2\Vert u\Vert _q^q-\lambda \Vert u\Vert _2^2\\&=\Vert u\Vert _p^p=\lim \limits _{k\rightarrow \infty }\Vert u_k\Vert _p^p\\&=\lim \limits _{k\rightarrow \infty }\left[ \Vert \nabla u_k\Vert _2^2-\mu \tau (\Vert \nabla u_k\Vert _2)\Vert u_k\Vert _q^q-\frac{\mu \tau '(\Vert \nabla u_k\Vert _2)}{q}\Vert \nabla u_k\Vert _2\Vert u_k\Vert _q^q-\lambda \Vert u_k\Vert _2^2 \right] \\&\ge A^2-\mu \tau (A)\Vert u\Vert _q^q-\frac{\mu \tau '(A)}{p}A\Vert u\Vert _q^q-\lambda \Vert u\Vert _2^2. \end{aligned} \end{aligned}$$

We claim that $1-\Vert u\Vert _q^q\frac{\mu \tau '(A)}{q}>0$. If not, then $q\le \mu \tau '(A)\Vert u\Vert _q^q$. From the properties of function $\tau$, we have the following two cases.

Case 1: If $A\in (0,R_0]\cup [R_1,+\infty )$ then $\tau '(A)=0$, we get a contradiction

$$\begin{aligned}0<q\le \mu \tau '(A)\Vert u\Vert _q^q=0.\end{aligned}$$

Case 2: If $A\in [R_0,R_1]$ then $\tau '(x)\in [0,\beta _{max})$, by Proposition 2.1 and Gagliardo-Nirenberg inequality (1.1) we have

$$\begin{aligned} \begin{aligned} q\le \mu \tau '(A)\Vert u\Vert _q^q&\le \mu \beta _{max}C_{N,q}^qa^{(1-\delta _q)q}\Vert \nabla u\Vert _2^{q\delta _q}\\&\le \mu \beta _{max}C_{N,q}^qa^{(1-\delta _q)q}R_2^{q\delta _q}\\&\le \mu \beta _{max}C_{N,q}^q \left( \frac{p}{2C_{N,p}^p}\right) ^{\frac{q\delta _q}{p\delta _p-2}}a^{-q\delta _q\frac{(1-\delta _p)p}{p\delta _p-2}} a^{(1-\delta _q)q}, \end{aligned} \end{aligned}$$

this contradicts condition (A.2).

Then we can deduce that $A=\Vert \nabla u\Vert _{2}$ and $\Vert u\Vert _2^2=a^2$. Up to a subsequence, $u_n \rightarrow u$ strongly in $H_r^1(\mathbb {R}^N)$. $\square$

Remark 3.1

When the formulas (A.1)) and (A.2) are satisfied, according to (3.7), Lemma 3.5 and 3.6 we know that the functional $\mathcal {J}_T$ has an unbounded sequence of pairs of radial solutions $\{(u_k,\lambda _k)\}\subset H^1(\mathbb {R}^N)\times \mathbb {R}^-$.

Proof of Theorem 1.1

From Remark 3.1, we know the functional $\mathcal {J}_T$ has an unbounded sequence of pairs of radial solutions $\{(u_k,\lambda _k)\}\subset H^1(\mathbb {R}^N)\times \mathbb {R}^-$, where $\Vert \nabla u_k\Vert _2^2\rightarrow \infty$ as $k\rightarrow \infty$. By Lemma 2.1 we can see that $\mathcal {J}_T(u)=\mathcal {J}_T(u)$ when $\Vert \nabla u\Vert _2^2\ge R_1$. Thus, we can obtain an unbounded subsequence, still denoted as $\{(u_k,\lambda _k)\}$, which is an unbounded sequence of pairs of radial solutionsan of the problem ($\mathcal {P}$). $\square$