Infinitely Many Solutions for Schrödinger–Poisson Systems and Schrödinger–Kirchhoff Equations

Abstract

By applying Clark’s theorem as altered by Liu and Wang and the truncation method, we obtain a sequence of solutions for a Schrödinger–Poisson system $\left\{\begin{array}{cc}-\Delta u+V\left(x\right)u+\varphi u=f\left(u\right)\hfill & \mathrm{in}{\mathbb{R}}^3,\hfill \\ -\Delta \varphi =u^2\hfill & \mathrm{in}{\mathbb{R}}^3\hfill \end{array}\right.$ with negative energy. A similar result is also obtained for the Schrödinger-Kirchhoff equation as follows:$-\left(1+{\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^2\right)\Delta u+V\left(x\right)u=f\left(u\right)u\in H^1\left({\mathbb{R}}^N\right)$.

1. Introduction

In this paper, we consider stationary Schrödinger–Poisson systems of the form

$$\left\{\begin{array}{cc}-\Delta u+V\left(x\right)u+\varphi u=f\left(u\right)\hfill & \mathrm{in}{\mathbb{R}}^3,\hfill \\ -\Delta \varphi =u^2\hfill & \mathrm{in}{\mathbb{R}}^3.\hfill \end{array}\right.$$

(1)

We assume the following conditions for the nonlinearity f and the potential V:

• $\left(f\right)\hspace{1em}f\in C\left(\mathbb{R}\right)$ is odd, for some $q\in \left(1,2\right)$ and $a>0$,

  $$\underset{t\to 0}{lim}{\displaystyle \frac{f\left(t\right)t}{{\left|t\right|}^q}}=a,\underset{\left|t\right|\to \infty }{lim}{\displaystyle \frac{f\left(t\right)t}{t^6}}=0;$$

  (2)
• $\left(V\right)\hspace{1em}V:{\mathbb{R}}^3\to \left(1,\infty \right)$ is continuous and $V^{-q/\left(2-q\right)}\in L\left({\mathbb{R}}^3\right)$.

The condition (2) means that the nonlinearity f is sublinear at the origin and subcritical at infinity. Under these mild conditions, we have the following multiplicity results.

Theorem 1.

Suppose $\left(f\right)$ and $\left(V\right)$ are satisfied, then the problem (1) has a sequence of weak solutions $\left\{\left(u_n,{\varphi }_n\right)\right\}\subset H^1\left({\mathbb{R}}^3\right)\times {\mathcal{D}}^{1,2}\left({\mathbb{R}}^3\right)$.

Example 1.

Let $f:\mathbb{R}\to \mathbb{R}$ and $V:{\mathbb{R}}^3\to \mathbb{R}$ be defined by

$$f\left(t\right)={\displaystyle \frac{t}{\sqrt{\left|t\right|}}}+{\left|t\right|}^{2}t,V\left(x\right)={(1+|x|}^2).$$

Then, f and V satisfy our assumptions $\left(f\right)$ and $\left(V\right)$ with $q=3/2$, respectively.

Remark 1.

Note that in the assumption $\left(V\right)$, we have assumed that V is positive. This condition can be replaced by the following weaker one which allows V to be negative somewhere:

• $(V*)\hspace{1em}V\in C\left({\mathbb{R}}^3\right)$, $V>0$ in ${\mathbb{R}}^3\setminus B_r$ and $V^{-q/\left(2-q\right)}\in L({\mathbb{R}}^3\setminus B_r)$ for some $r>0$.

To see this, set $\tilde{V}\left(x\right)=V\left(x\right)+m$, $\tilde{f}\left(t\right)=f\left(t\right)+mt$ with

$$m=1+\underset{B_r}{sup}\left|V\right|.$$

Then, $\tilde{f}$ and $\tilde{V}$ satisfy the assumptions $\left(f\right)$ and $\left(V\right)$, and hence we can apply Theorem 1 to the equivalent problem

$$\left\{\begin{array}{cc}-\Delta u+\tilde{V}\left(x\right)u+\varphi u=\tilde{f}\left(u\right)\hfill & in{\mathbb{R}}^3,\hfill \\ -\Delta \varphi =u^2\hfill & in{\mathbb{R}}^3.\hfill \end{array}\right.$$

The problem (1) is variational. Let ${\varphi }_u$ be ths solution of the second equation in (1). As proposed by Benci et al. [1,2], it is well known that if u is a critical point of

$$\Phi \left(u\right)={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^3}\left({|\nabla u|}^2+V\left(x\right)u^2\right)+{\displaystyle \frac{1}{4}}{\int }_{{\mathbb{R}}^3}{\varphi }_u{u}^2-{\int }_{{\mathbb{R}}^3}F\left(u\right)$$

(3)

being $F\left(t\right)={\int }_0^{t}f$, then $\left(u,{\varphi }_u\right)$ is a solution of (1). Using this idea, many results on (1) have been obtained assuming that

$$\begin{array}{c}\hfill \underset{t\to 0}{lim}{\displaystyle \frac{f\left(t\right)}{t}}=0\end{array}$$

(4)

in the last two decades; see [3,4,5] for a case where the Schrödinger operator $S=-\Delta +V$ is positive and [6] for a case where the Schrödinger operator S is indefinite.

In all these papers, and many papers on Schrödinger–Poisson systems, some conditions on the nonlinearity f, like the Ambrosetti–Rabinowitz condition, are needed to ensure that the variational functional $\Phi$ satisfies the Palais–Smale $\left(PS\right)$ condition.

Unlike all these papers, our assumption $\left(f\right)$ on the nonlinearity f implies that the limit in (4) is infinity, and our assumptions on f are not sufficient for ensuring the boundedness of $\left(PS\right)$ sequences. Motivated by He and Wu [7], who studied the semilinear elliptic boundary value problem

$$\left\{\begin{array}{cc}-\Delta u+V\left(x\right)u=f(x,u)\hfill & \mathrm{in}\Omega ,\hfill \\ u=0\hfill & \mathrm{on}\partial \Omega \hfill \end{array}\right.$$

on a bounded domain $\Omega \subset {\mathbb{R}}^N$, we apply the truncation method and a version of Clark’s theorem created by Liu and Wang [8] to overcome this difficulty.

Using the same idea, we obtained a similar result for Schrödinger–Kirchhoff equations of the form ($N\ge 3$)

$$-\left(1+{\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^2\right)\Delta u+V\left(x\right)u=f\left(u\right)u\in H^1\left({\mathbb{R}}^N\right).$$

(5)

We make the following assumptions:

• $\left(f^N\right)\hspace{1em}$$f\in C\left(\mathbb{R}\right)$ is odd, for some $q\in \left(1,2\right)$ and $a>0$,

  $$\underset{t\to 0}{lim}{\displaystyle \frac{f\left(t\right)t}{{\left|t\right|}^q}}=a,\underset{\left|t\right|\to \infty }{lim}{\displaystyle \frac{f\left(t\right)t}{t^{2^{*}}}}=0;$$
• $\left(V^N\right)\hspace{1em}$$V:{\mathbb{R}}^N\to \left(1,\infty \right)$ is continuous and $V^{-q/\left(2-q\right)}\in L\left({\mathbb{R}}^N\right)$.

Note that $2^{*}=2N/(N-2)$ is the critical Sobolev exponent. If $N=3$, then $2^{*}=6$ and $\left(f^N\right)$ and $\left(V^N\right)$ reduce to $\left(f\right)$ and $\left(V\right)$, respectively.

Theorem 2.

Suppose $\left(f^N\right)$ and $\left(V^N\right)$ are satisfied, then the problem (5) has a sequence of weak solutions $\left\{u_n\right\}\subset H^1\left({\mathbb{R}}^N\right)$.

For problem (5), we make a remark similar to Remark 1. This paper is organized as follows. In Section 2, we present the functional spaces as our frameworks for studying problems (1) and (5), and Clark’s theorem as altered by Liu and Wang [8] is also recalled in this section. The proofs of Theorems 1 and 2 will be given in Section 3 and Section 4, respectively.

2. Preliminaries

Equip the subspace ($N\ge 3$)

$$X^N=\left\{u\in H^1\left({\mathbb{R}}^N\right)\left|{\int }_{{\mathbb{R}}^N}V\left(x\right)u^2<\infty \right.\right\}$$

with the norm $\parallel ·\parallel$ and the corresponding inner product $\left(·,·\right)$ given by

$$∥u∥={\left({\int }_{{\mathbb{R}}^N}\left({\left|\nabla u\right|}^2+V\left(x\right)u^2\right)\right)}^{1/2},(u,v)={\int }_{{\mathbb{R}}^N}\left(\nabla u·\nabla v+V\left(x\right)uv\right)u,v\in X^N.$$

Then, $X^N$ is a Hilbert space. Since $C_0^{\infty }\left({\mathbb{R}}^3\right)\subset X^3$, to prove Theorem 1, it suffices to find a sequence of critical points of the $C^1$-functional $\Phi :X^3\to \mathbb{R}$ given in (3). Note that by the definition of $∥·∥$,

$$\Phi \left(u\right)={\displaystyle \frac{1}{2}}{\parallel u\parallel }^2+{\displaystyle \frac{1}{4}}{\int }_{{\mathbb{R}}^3}{\varphi }_u{u}^2-{\int }_{{\mathbb{R}}^3}F\left(u\right).$$

Obviously, the embedding $X^N\hookrightarrow H^1\left({\mathbb{R}}^N\right)$ is continuous. According to [9] [Lemma 2.1], we obtain the following result.

Proposition 1.

Under the assumption $\left(V^N\right)$, $X^N$ can be continuously embedded into $L^s\left({\mathbb{R}}^N\right)$ for $s\in \left[q,2^{*}\right]$ and the embedding is compact for $s\in [q,2^{*})$.

Corollary 1.

Under the assumptions $\left(f^N\right)$ and $\left(V^N\right)$, if $u_n\rightharpoonup u$ in $X^N$, then up to a subsequence

$${\int }_{{\mathbb{R}}^N}f\left(u_n\right)(u_n-u)\to 0.$$

(6)

Proof. 

Given $\epsilon >0$, there is $C_{\epsilon }>0$ such that

$$\left|f\left(t\right)\right|\le C_{\epsilon }{\left|t\right|}^{q-1}+\epsilon {\left|t\right|}^{2^{*}-1}.$$

Thus, by Hölder inequality

$$\begin{array}{cc}\hfill \left|{\int }_{{\mathbb{R}}^N}f\left(u_n\right)(u_n-u)\right|& \le C_{\epsilon }{\int }_{{\mathbb{R}}^N}|u_n{|}^{q-1}|u_n-u|+\epsilon {\int }_{{\mathbb{R}}^N}{\left|u_n\right|}^{2^{*}-1}\left|u_n-u\right|\hfill \\ \hfill & \le C_{\epsilon }{\left|u_n\right|}_q^{q-1}{\left|u_n-u\right|}_q+\epsilon {\left|u_n\right|}_{2^{*}}^{2^{*}-1}{\left|u_n-u\right|}_{2^{*}}.\hfill \end{array}$$

Because $\left\{u_n\right\}$ is bounded in $L^{2^{*}}\left({\mathbb{R}}^N\right)$ and $u_n\to u$ in $L^q\left({\mathbb{R}}^N\right)$ by Proposition 1, we obtain

$$\underset{n\to \infty }{\overline{lim}}\left|{\int }_{{\mathbb{R}}^N}f\left(u_n\right)(u_n-u)\right|\le C\epsilon$$

for some $C>0$, which implies (6). □

To find the critical points of $\Phi$, the properties of ${\varphi }_u$ and the functional

$$\mathcal{N}\left(u\right)={\displaystyle \frac{1}{4}}{\int }_{{\mathbb{R}}^3}{\varphi }_u{u}^2$$

(7)

are crucial. Similar to [10] [Lemma 2.2 (1)], we have the following proposition.

Proposition 2.

Let ${\varphi }_u\in D^{1,2}\left({\mathbb{R}}^3\right)$ be the unique solution of $-\Delta \varphi =u^2$ for $u\in X^3$. Then, there is a constant $b>0$ such that

$${∥{\varphi }_u∥}_{{\mathcal{D}}^{1,2}}\le {b\left|u\right|}_{12/5}^2,\left|{\displaystyle \frac{1}{4}}{\int }_{{\mathbb{R}}^3}{\varphi }_u{u}^2\right|\le b{\parallel u\parallel }^4.$$

To conclude this section, we recall Clark’s theorem, which will be needed for proving our main results.

Proposition 3

([8] [Theorem 1.1]). Let X be a Banach space and $\Psi \in C^1(X,\mathbb{R})$ be an even coercive functional satisfying the ${\left(PS\right)}_c$ condition for $c\le 0$ and $\Psi \left(0\right)=0$. If for any $k\in \mathbb{N}$ there is a k-dimensional subspace $W_k$ and ${\delta }_k>0$ such that

$$\underset{W_k\cap S_{{\delta }_k}}{sup}\Psi <0,$$

(8)

where for $r>0$, $S_r=\left\{u\in W\mid ∥u∥=r\right\}$, then Ψ has a sequence of critical points $u_k\ne 0$ such that $\Psi \left(u_k\right)\le 0$, $u_k\to 0$.

3. Proof of Theorem 1

Let $\eta :[0,\infty )\to \left[0,1\right]$ be a decreasing $C^{\infty }$-function such that $\left|{\eta }^{\prime }\left(t\right)\right|\le 2$,

$$\eta \left(t\right)=1\mathrm{for}t\in \left[0,1\right],\eta \left(t\right)=0\mathrm{for}t\in [2,\infty ).$$

We consider the truncated functional $\Psi :X^3\to \mathbb{R}$,

$$\Psi \left(u\right)={\displaystyle \frac{1}{2}}{\parallel u\parallel }^2+{\displaystyle \frac{1}{4}}{\int }_{{\mathbb{R}}^3}{\varphi }_u{u}^2-\eta \left({∥u∥}^2\right){\int }_{{\mathbb{R}}^3}F\left(u\right).$$

The derivative of $\Psi$ is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \langle {\Psi }^{\prime }\left(u\right),h\rangle & =\left(u,h\right)+{\int }_{{\mathbb{R}}^3}{\varphi }_{u}uh-\eta \left({∥u∥}^2\right){\int }_{{\mathbb{R}}^3}f\left(u\right)h-\left({\int }_{{\mathbb{R}}^3}F\left(u\right)\right){\eta }^{\prime }\left({∥u∥}^2\right)\left(u,h\right)\hfill \\ \hfill & =\left(1-\left({\int }_{{\mathbb{R}}^3}F\left(u\right)\right){\eta }^{\prime }\left({∥u∥}^2\right)\right)\left(u,h\right)+{\int }_{{\mathbb{R}}^3}{\varphi }_{u}uh-\eta \left({∥u∥}^2\right){\int }_{{\mathbb{R}}^3}f\left(u\right)h\hfill \end{array}\end{array}$$

(9)

for $u,h\in X^3$.

Lemma 1.

The functional Ψ is coercive and satisfies ${\left(PS\right)}_c$ for $c\le 0$.

Proof. 

Let $\left\{u_n\right\}\subset X^3$ be a sequence such that $∥u_n∥\to \infty$. Then, for n large, we have $∥u_n∥>1$ and $\eta \left({∥u∥}^2\right)=0$. Thus

$$\Psi \left(u_n\right)={\displaystyle \frac{1}{2}}{∥u_n∥}^2+{\displaystyle \frac{1}{4}}{\int }_{{\mathbb{R}}^3}{\varphi }_{u_n}{u}_n^2\ge {\displaystyle \frac{1}{2}}{∥u_n∥}^2\to +\infty .$$

Hence $\Psi$ is coercive.

For $c\le 0$, let $\left\{u_n\right\}\subset X^3$ be a ${\left(PS\right)}_c$ sequence of $\Psi$. That is $\Psi \left(u_n\right)\to c$, ${\Psi }^{\prime }\left(u_n\right)\to 0$. Then for n large we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill -\eta \left({∥u_n∥}^2\right){\int }_{{\mathbb{R}}^3}F\left(u_n\right)& =\Psi \left(u_n\right)-{\displaystyle \frac{1}{2}}{∥u_n∥}^2-{\displaystyle \frac{1}{4}}{\int }_{{\mathbb{R}}^3}{\varphi }_{u_n}{u}_n^2\hfill \\ \hfill & \le \Psi \left(u_n\right)-{\displaystyle \frac{1}{2}}{∥u_n∥}^2.\hfill \end{array}\end{array}$$

(10)

We claim that

$$1-\left({\int }_{{\mathbb{R}}^3}F\left(u_n\right)\right){\eta }^{\prime }\left({∥u_n∥}^2\right)\ge 1.$$

(11)

We consider two cases:

• If $∥u_n∥<1$, then ${\eta }^{\prime }\left({∥u_n∥}^2\right)=0$; hence (11) is true.
• If $∥u_n∥\ge 1$, then the right-hand side of (10) is negative or n large. Thus

$${\int }_{{\mathbb{R}}^3}F\left(u_n\right)\ge 0$$

which implies (11) because ${\eta }^{\prime }\left({∥u_n∥}^2\right)\le 0$.

Because $\Psi$ is coercive, the ${\left(PS\right)}_c$ sequence $\left\{u_n\right\}$ is bounded in X. Thus, up to a subsequence, we have

$$u_n\rightharpoonup u\mathrm{in}{X}^3,u_n\to u\mathrm{in}{L}^{12/5}\left({\mathbb{R}}^3\right).$$

By Hölder inequality and Proposition 2, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|{\int }_{{\mathbb{R}}^3}{\varphi }_{u_n}{u}_n(u_n-u)\right|& \le {\left|{\varphi }_{u_n}\right|}_6{\left|u_n\right|}_{12/5}{\left|u_n-u\right|}_{12/5}\hfill \\ \hfill & \le S{∥{\varphi }_{u_n}∥}_{{\mathcal{D}}^{1,2}}{\left|u_n\right|}_{12/5}{\left|u_n-u\right|}_{12/5}\hfill \\ \hfill & \le Sb{\left|u_n\right|}_{12/5}^2{\left|u_n-u\right|}_{12/5}\to 0,\hfill \end{array}\end{array}$$

(12)

where S is the best constant of the embedding ${\mathcal{D}}^{1,2}\left({\mathbb{R}}^3\right)\to L^6\left({\mathbb{R}}^3\right)$. Consequently, from (9), (11), Corollary 1 and ${\left|\eta \right|}_{\infty }\le 1$ we deduce

$$\begin{array}{cc}\hfill \left|\left(u_n,u_n-u\right)\right|& \le \left(1-\left({\int }_{{\mathbb{R}}^3}F\left(u_n\right)\right){\eta }^{\prime }\left({∥u_n∥}^2\right)\right)\left|\left(u_n,u_n-u\right)\right|\hfill \\ \hfill & \le \left|\langle {\Psi }^{\prime }\left(u_n\right),u_n-u\rangle \right|+\eta \left({∥u_n∥}^2\right)\left|{\int }_{{\mathbb{R}}^3}f\left(u_n\right)(u_n-u)\right|\hfill \\ \hfill & +\left|{\int }_{{\mathbb{R}}^3}{\varphi }_{u_n}{u}_n(u_n-u)\right|\to 0.\hfill \end{array}$$

We conclude that $∥u_n∥\to ∥u∥$. Noting $u_n\rightharpoonup u$ in $X^3$, we obtain $u_n\to u$ in $X^3$. □

Now we are ready to prove Theorem 1.

Proof of Theorem 1.

Since $\Psi \left(u\right)=\Phi \left(u\right)$ for $u\in B_1$ (the unit ball in $X^3$), it suffices to show that $\Psi$ has a sequence of critical points $\left\{u_k\right\}\subset X^3$ satisfying $u_k\to 0$. For this purpose, we shall apply Proposition 3.

Firstly, we remark that by condition $\left(f\right)$, there is $A>0$ such that

$$F\left(t\right)\ge {A\left|t\right|}^q\mathrm{for}\left|t\right|\le 1.$$

(13)

Let $k\in \mathbb{N}$ and $W_k$ be a k-dimensional subspace of $X^3$. Since all norms on $W_k$ are equivalent, there is $\delta \in \left(0,1\right)$ such that $∥u∥\le \delta$ implies ${\left|u\right|}_{\infty }\le 1$; thus

$${\int }_{{\mathbb{R}}^3}F\left(u\right)\ge A{\left|u\right|}_q^q$$

for $u\in W_k\cap B_{\delta }$. Hence, for $u\in W_k\cap B_{\delta }$, using Proposition 2 we obtain

$$\begin{array}{cc}\hfill \Psi \left(u\right)& ={\displaystyle \frac{1}{2}}{\parallel u\parallel }^2+{\displaystyle \frac{1}{4}}{\int }_{{\mathbb{R}}^3}{\varphi }_u{u}^2-{\int }_{{\mathbb{R}}^3}F\left(u\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\parallel u\parallel }^2+{b\parallel u\parallel }^4-A{\left|u\right|}_q^q\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\parallel u\parallel }^2+{b\parallel u\parallel }^4-C{\parallel u\parallel }^q=\gamma \left(∥u∥\right)\hfill \end{array}$$

(14)

being $\gamma \left(t\right)={\displaystyle \frac{1}{2}}{t}^2+b{t}^4-C{t}^q$ for some constant $C>0$. Because $q\in \left(1,2\right),$ it is clear that there is ${\delta }_k\in \left(0,\delta \right)$ such that $\gamma \left({\delta }_k\right)<0$. Hence,

$$\underset{W_k\cap S_{{\delta }_k}}{sup}\Psi \le \gamma \left({\delta }_k\right)<0$$

(15)

As we have seen, the $C^1$-functional $\Psi$ is even, coercive and satisfies ${\left(PS\right)}_c$ for $c\le 0$. Since $\Psi \left(0\right)=0$ is trivially true, by Proposition 3 $\Psi$ has a sequence of critical points converging to the zero function. This completes the proof of Theorem 1. □

4. Proof of Theorem 2

Proof of Theorem 2.

Now we will prove Theorem 2 using similar method. The solutions of (5) can be found as critical points of $\phi :X^N\to \mathbb{R}$ given by

$$\phi \left(u\right)={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^N}\left({\left|\nabla u\right|}^2+V\left(x\right)u^2\right)+{\displaystyle \frac{1}{4}}{\left({\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^2\right)}^2-{\int }_{{\mathbb{R}}^N}F\left(u\right).$$

As before, to overcome the difficulty that $\left(PS\right)$ sequences may be unbounded, we consider the truncated functional $\psi :X^N\to \mathbb{R}$,

$$\psi \left(u\right)={\displaystyle \frac{1}{2}}{∥u∥}^2+{\displaystyle \frac{1}{4}}{\left({\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^2\right)}^2-\eta \left({∥u∥}^2\right){\int }_{{\mathbb{R}}^N}F\left(u\right),$$

where $\eta$ is the function given at the beginning of §3. Then,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \langle {\psi }^{\prime }\left(u\right),h\rangle & =\left(u,h\right)+\left({\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^2\right){\int }_{{\mathbb{R}}^N}\nabla u·\nabla h-\eta \left({∥u∥}^2\right){\int }_{{\mathbb{R}}^N}f\left(u\right)h\hfill \\ \hfill & -\left({\int }_{{\mathbb{R}}^N}F\left(u\right)\right){\eta }^{\prime }\left({∥u∥}^2\right)\left(u,h\right)\hfill \\ \hfill & =\left(1-\left({\int }_{{\mathbb{R}}^N}F\left(u\right)\right){\eta }^{\prime }\left({∥u∥}^2\right)\right)\left(u,h\right)\hfill \\ \hfill & +\left({\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^2\right){\int }_{{\mathbb{R}}^N}\nabla u·\nabla h-\eta \left({∥u∥}^2\right){\int }_{{\mathbb{R}}^N}f\left(u\right)h.\hfill \end{array}\end{array}$$

(16)

Similar to the proof of Lemma 1, the functional $\psi :X^N\to \mathbb{R}$ is coercive.

Given $c\le 0$, let $\left\{u_n\right\}\subset X^N$ be a ${\left(PS\right)}_c$ sequence. Then $\left\{u_n\right\}$ is bounded, so

$$u_n\rightharpoonup u\mathrm{in}{X}^N$$

for some $u\in X^N$, and we have an analog of (11):

$$1-\left({\int }_{{\mathbb{R}}^N}F\left(u_n\right)\right){\eta }^{\prime }\left({∥u_n∥}^2\right)\ge 1.$$

(17)

However, unlike in the proof of Lemma 1, we could not apply Proposition 2 to obtain the analog of (12):

$$\left({\int }_{{\mathbb{R}}^N}{\left|\nabla u_n\right|}^2\right){\int }_{{\mathbb{R}}^N}\nabla u_n·\nabla (u_n-u)\to 0.$$

To go around this difficulty, by the boundedness of $\left\{u_n\right\}$ in $X^N$ we can observe that the left-hand side of (17) is bounded, and hence up to a subsequence we may assume

$$1-\left({\int }_{{\mathbb{R}}^N}F\left(u_n\right)\right){\eta }^{\prime }\left({∥u_n∥}^2\right)\to \alpha \ge 1,{\int }_{{\mathbb{R}}^N}{\left|\nabla u_n\right|}^2\to \beta \ge 0.$$

Now, by analyzing $〈{\psi }^{\prime }\left(u_n\right),u_n-u〉$ using (16)

$$\underset{n\to \infty }{\underline{lim}}{\int }_{{\mathbb{R}}^N}\nabla u_n·\nabla (u_n-u)=\left(\underset{n\to \infty }{\underline{lim}}{\left|\nabla u_n\right|}_2^2\right)-{\left|\nabla u\right|}_2^2\ge 0,$$

and

$${\int }_{{\mathbb{R}}^N}f\left(u_n\right)(u_n-u)\to 0,$$

(see Corollary 1) we obtain

$$\begin{array}{cc}\hfill & \alpha \underset{n\to \infty }{\overline{lim}}\left({∥u_n∥}^2-{∥u∥}^2\right)=\underset{n\to \infty }{\overline{lim}}\left(1-\left({\int }_{{\mathbb{R}}^N}F\left(u_n\right)\right){\eta }^{\prime }\left({∥u_n∥}^2\right)\right)\left(u_n,u_n-u\right)\hfill \\ \hfill & =\underset{n\to \infty }{\overline{lim}}\left(\langle {\psi }^{\prime }\left(u_n\right),u_n-u\rangle -\left({\int }_{{\mathbb{R}}^N}{\left|\nabla u_n\right|}^2\right){\int }_{{\mathbb{R}}^N}\nabla u_n·\nabla (u_n-u)\right.\hfill \\ \hfill & \left.+\eta \left({∥u_n∥}^2\right){\int }_{{\mathbb{R}}^N}f\left(u_n\right)(u_n-u)\right)\hfill \\ \hfill & =-\beta \underline{lim}{\int }_{{\mathbb{R}}^N}\nabla u_n·\nabla (u_n-u)\le 0.\hfill \end{array}$$

Thus,

$$\underset{n\to \infty }{\overline{lim}}\parallel u_n\parallel \le \parallel u\parallel ,$$

which implies that $∥u_n∥\to ∥u∥$, and we still obtain $u_n\to u$ in $X^N$. This completes the proof that $\psi$ satisfies ${\left(PS\right)}_c$ for $c\le 0$.

Similar to (14) in the proof of Theorem 1, we can prove that $\psi$ satisfies the geometric assumption (8) in Proposition 3. Hence, we may apply Proposition 3 to obtain a sequence of critical points $\left\{u_k\right\}$ of $\psi$ with $∥u_k∥\to 0$. For k large, all $u_k$ are critical points of $\phi$ because $\psi \left(u\right)=\phi \left(u\right)$ for $u\in B_1$. Therefore, Theorem 2 is proved. □