Multiplicity of solutions for Schrödinger-Poisson system with critical exponent in $\mathbb{R}^{3}$

1.   Introduction

In this article, we are devoted to the following Schrödinger-Poisson system

where $\lambda > 0$ is a parameter, $1 < p < 2$. To state our results, we impose some conditions on $h$ and $k$ as follows:

$(A_{1})\quad h\in L^{\frac{6}{6-p}}(\mathbb{R}^{3}), \, h(x)\geq0$ and $h(x)\not\equiv 0.$

$(A_{2})\quad k\in L^{2}(\mathbb{R}^{3}), \; k(x)\geq0$ and $k(x)\not\equiv0.$

$(A_{3})\quad s\in C(\mathbb{R}^{3})\cap L^\infty(\mathbb{R}^{3}), \; s(x) > 1.$

$(A_{4})\quad s\in C(\mathbb{R}^{3})\cap L^\infty(\mathbb{R}^{3}), \; s(0) = 0, \; s(x) > 0\; a.e.\; \text{in}\; \mathbb{R}^3\; \text{and} \lim\limits_{|x|\rightarrow\infty}s(x) = 0.$

As we all know the Schrödinger-Poisson system has a strong physical meaning due to the influence in quantum mechanics models (see e.g. ^[5,15]) and in semiconductor theory (see e.g. ^[18,19]). The crucial tools to study the existence and multiplicity of solutions about nonlinear differential equations are the variational method and the critical point theory (see e.g. ^[2,26]). From an academic point of view, these methods present an interesting competition between local and nonlocal nonlinearities. Problem (1.1) is derived from the following Schrödinger-Poisson system

which is also called the Schrödinger-Maxwell equation, was firstly introduced in ^[4] while describing the interacting between solitary waves and an electrostatic field in quantum mechanics. While studying the Schrödinger-Poisson system, one has to face many obstacles since the existence of the non-local term, especially in the critical case, the invariance by dilations of $\mathbb{R}^3$ makes the problems much harder to deal with. In the past few years, a number of papers are devoted to the existence of solutions for (1.2) under various assumptions on $V$, $k$ and $f$. In ^[8], D'Aprile and Mugnai firstly proved the existence results in the subcritical case. And the first non-existence result was given in ^[9] for the critical case. After that, Ruiz in ^[21] obtained more existence results and properties of the non-local term $\phi$. Based on the work of ^[21], Azzollini and Pomponio ^[3] obtained the existence of ground state solutions for (1.2) where $f(x, u) = |u|^{p-1}u$ with $2 < p < 5$ when $V$ is a positive constant and $3 < p < 5$ when $V$ is a non-constant potential. After that, Zhang, Ma and Xie ^[28] studied the following problem with critical exponent

where $V\in L^\frac{3}{2}(\mathbb{R}^3)$, and proved the existence of bound state solutions.

In recent years, some researchers are interested in the existence of solutions involving concave-convex nonlinearities. For example, Zhang ^[27] obtained ground state and nodal solutions of following problem with critical exponent

where $p\in(4, 6).$ Later, in ^[14], Li and Tang considered the following Schrödinger-Poisson system with negative coefficient of nonlocal term

they proved Problem (1.5) possesses at least two solutions by Mountain Pass Theorem and Ekeland's Variational Principle. As for more results treating this problem or similar one, readers can refer to ^{[1,7,10,13,22,23,24]} and references therein.

All above works are to study existence of solutions of the Schrödinger-Poisson system under different conditions. Here, we have to highlight the fact that one of the main attentions of interest in our present paper is to prove the existence of infinitely many solutions. To the best of our knowledge, it seems that there are no results about infinitely many solutions while concerning negative coefficient nonlocal term. The first purpose in our paper is to establish the multiplicity of solutions possessing negative energy of the problem (1.1). Furthermore, we are also devote to studying the convergent properties of energy corresponding to the solutions. The critical exponential growth makes the problem complicated due to the lack of compactness, thus we use the concentration compactness principle to restore compactness. And we will introduce a cut-off functional which is bounded from below, by analyzing the properties of the cut-off functional, utilizing $Z_{2}$ index theory, we can obtain the first result. To demonstrate our second result, we assume some extra conditions on $k, \; h\; \text{and}\; s$, by using Fountain Theorem, we prove the existence of the multiple solutions possessing positive energy.

Next, we will state our main results.

Theorem 1.1. Suppose $1 < p < 2$, the hypotheses $(A_{1}), (A_{2})\; and\; (A_{3})$ hold. If $h(x) > 0$ is bounded on some open subset $\Omega\subset\mathbb{R}^{3}$ with $|\Omega| > 0$. Then there exists $\lambda^{\ast} > 0$ such that for all $\lambda\in(0, \lambda^{\ast})$, Problem (1.1) has infinitely many solutions with negative energy. Moreover, there exists a sequence of the critical values corresponding to the solutions which converges to zero.

In order to give the second result, we need to introduce some notations. Denote $O(3)$ to be the group of orthogonal linear transformations in $\mathbb{R}^3$ and let $T\subset O(3)$ be a subgroup. Set $|T|: = \inf \limits_{x\in \mathbb{R}^3, x\neq0}|T_x|$, where $T_x: = \{\tau x:\tau\in O(3)\}$ for $x\neq0$. Moreover, a function $f: \mathbb{R}^3\rightarrow\mathbb{R}$ is called $T$-invariant if $f(\tau x) = f(x)$ for all $\tau \in T$ and $x\in \mathbb{R}^3.$

Theorem 1.2. Suppose $1 < p < 2$, the hypotheses $(A_{1}), (A_{2})$ and $(A_{4})$ hold. Assume $k(x), \; h(x)$ and $s(x)$ are $T$-invariant. Moreover, let $|T| = \infty$. Then Problem (1.1) has infinitely many solutions with positive energy.

Remark 1.1. The results obtained in our paper extend the ones in ^[14]. To be more precise, the authors ^[14] obtained just two solutions. Here, by the argument of $Z_2$ index theory, we prove the existence of infinitely many small solutions with negative energy, besides, we also obtain a sequence of high energy solutions by Fountain Theorem.

This paper is organized as follows. In section 2, we give some notations and preliminaries, for the readers' convenience, we also describe the main mathematical tools which we shall use. In section 3, we prove Theorem 1.1 by the truncated technique. Section 4 is devoted to the proof of Theorem 1.2.

2.   Preliminaries and variational setting

Hereafter we use the following notations.

$L^{s}: = L^{s}(\mathbb{R}^{3})\; (1\leq s < \infty)$ is the usual Lebesgue space with the norm defined by

$||\cdot||_{\infty}$ denotes the $L^{\infty}$-norm and $D^{1, 2}: = D^{1, 2}(\mathbb{R}^{3}) = \{u\in L^{6}(\mathbb{R}^{3})|\; \nabla u\in L^{2}(\mathbb{R}^{3})\}$ with the norm defined by

For any $\rho > 0$ and $z\in \mathbb{R}^{3}$, $B_{\rho}(z)$ denotes the ball of radius $\rho$ centered at $z$, and $|B_{\rho}(z)|$ denotes its Lebesgue measure. $C, \; \hat{C}, \; C_{p}, \; C_{1}, \; C_{2}, \cdots$are various positive constants which can change from line to line.

We now recall some known results. For all $u\in D^{1, 2}$, the linear functional $Lu$ is defined by

By $(A_{2})$, Hölder and Sobolev inequalities, we obtain

Thanks to the Lax-Milgram theorem, for every $u\in D^{1, 2}$, the Poisson equation

exists a unique solution $\phi_{u}\in D^{1, 2}$ and

It is easy to see that $\phi_{u}$ satisfies

for any $v\in D^{1, 2}$. Furthermore, by (2.1), (2.2), Hölder and Sobolev inequalities, the relations

hold, where $S$ is the best Sobolev constant defined by

Substituting $\phi_{u}$ into (1.1), we get

It is standard to see that the solutions of (1.1) are the critical points of the functional defined by

for $u\in D^{1, 2}.$ Hence, we just say that $u\in D^{1, 2}$, instead of $(u, \phi_{u})\in D^{1, 2} \times D^{1, 2}$, is a weak solution of system (1.1). It is easy to see that $I(u)\in C^{1}(D^{1, 2}, \mathbb{R})$ and

for all $\varphi\in D^{1, 2}.$

Now we define the operator

and set

In the following lemma, we conclude some properties of $\Phi\,$ which are useful for studying our problems.

Lemma 2.1. (^[21])

1. $\Phi$ is continuous;

2. $\Phi$ maps bounded sets into bounded sets;

3. $\Phi(t u) = t^{2}\Phi(u)$ for all $t \in \mathbb{R}$;

4. If $u_{n}\rightharpoonup u\in D^{1, 2}$, then $\Phi(u_{n})\rightarrow\Phi(u)$ in $D^{1, 2}$;

5. If $u_{n}\rightharpoonup u\in D^{1, 2}$, then $N(u_n)\rightarrow N(u)$, as $n\rightarrow\infty$.

Definition 2.2. Let $Y$ be a Banach space and $I: Y\rightarrow \mathbb{R}$ be a differentiable functional. A sequence $\{u_{k}\}\subset Y$ is called a $(PS)_c$ sequence for $I$ if $I(u_{k})\rightarrow c$ and $I'(u_{k})\rightarrow0$ as $k\rightarrow\infty.$ If every $(PS)_c\,$ sequence for $I$ has a converging subsequence $(in\; Y)$, we say that $I$ satisfies the $(PS)_c$ condition.

Lemma 2.3. Assume that $(A_{1})$, $(A_{2})$ and $(A_{3})$ hold. Let $\{u_{n}\}\subset D^{1, 2}$ be a $(PS)_{c}$ sequence for $I$, then $\{u_{n}\}$ is bounded in $D^{1, 2}$. Moreover, if $c < 0$, there exists $\lambda^{**} > 0$ such that $I$ satisfies the $(PS)_{c}$ condition for all $\lambda\in(0, \lambda^{**})$.

Proof. Since $\{u_{n}\}$ is a $(PS)_{c}$ sequence, we have

On one hand, by (2.4), we can easily get

On the other hand, we have

By $(A_{1})$, Sobolev and Hölder inequalities, we find

In view of (2.5)–(2.7), we get

Since $1 < p < 2$, we obtain that $\{u_{n}\}$ is bounded in $D^{1, 2}$. Thus there exists a subsequence, still denoted by $\{u_{{n}}\}$, and $u\in {D^{1, 2}}$, such that

Moreover, we get $|u_{n}|^{p}\rightharpoonup |u|^{p}$ in $L^\frac{6}{p}$ (see Proposition 4.7.12 in ^[6]). By $(A_{1})$, we can conclude that

Next, we want to use the concentration compactness principle to restore the compactness. Using the fact that $\{u_{n}\}$ is bounded in $D^{1, 2}$, by the concentration compactness principle in ^[16,17], we may suppose there exists a subsequence, still denoted by $\{u_{n}\},$ such that

where $\mu, \; \mu_i, \; \nu$ and $\nu_i$ are nonnegative measures, $\Gamma$ is an at most countable index set, $\{a_{i}\}\subset \mathbb{R}^{3}$ is a sequence and $\delta_{a_{i}}$ is the Dirac mass at $a_{i}$. Moreover, we have

where $S$ is given in (2.3).

We claim that $\Gamma$ is empty. Indeed, if $\Gamma$ is not empty, then there exists $i\in \Gamma$ such that $\mu _{i}\neq0.$ For $\varepsilon > 0$ small, we introduce a cut-off function centered at $a_{i}$ as following

and $0\leq\varphi^{i}_{\varepsilon}(x)\leq1$, $|\nabla\varphi^{i}_{\varepsilon}(x)|\leq\frac{4}{\varepsilon}$. By (2.4) we can obtain

which implies

Step 1. We prove $\nu_{i}^{\frac{1}{3}}\geq \sqrt{\frac{S}{s(a_i)}}$.

Since $\{u_n\}$ is bounded, using Hölder inequality, we can obtain

where $B_{\varepsilon}(a_{i}) = \{x\in \mathbb{R}^{3}|\; \; |x-a_{i}| < \varepsilon\}$. By (2.10), we get

and

In view of (2.12)-(2.17), we get $\mu_{i}\leq s(a_i)\nu_{i}$. By (2.11) we obtain

Step 2. We prove our claim.

Let $\varphi_{R}(x)$ be a cut-off function which satisfies

and $0\leq\varphi_{R}(x)\leq1$, $|\nabla\varphi_{R}(x)| < \frac{2}{R}$. By (2.5) and (2.9), we obtain

Using Hölder and Young inequalities ($\varepsilon$ small enough), we obtain

Hence we have

Choose $\lambda _{1}$ small enough such that $\frac{1}{4}\mu_{i}+\frac{1}{12}s(a_i)\nu_{i}-C_\varepsilon\lambda^{\frac{6}{6-p}} > 0$ for all $\lambda\in (0, \lambda_{1})$, which contradicts to $c < 0$. Thus $\Gamma$ is empty.

By the claim, we get

which implies

We define

and

From ^[16], we know that $\nu_{\infty}\; \text{and}\; \mu _{\infty}$ satisfy

where $\mu$ and $\nu$ are the same as above.

In the following discussion, we want to prove $\mu_{\infty} = \nu_{\infty} = 0$. Let $\eta_{R}\in C^{1}(\mathbb{R}^{3})$ be such that

with $0\leq\eta_{R}(x)\leq1$ and $|\nabla\eta_{R}(x)| < \frac{2}{R}$. From (2.4), we get

which gives

Since $\{u_{n}\}$ is bounded, by Hölder inequality, we have

Moreover, by Lemma 2.1, (2.9) and the definitions of $\mu_{\infty}$ and $\nu_{\infty}$, we obtain

and

Therefore, by (2.24)–(2.29), we obtain

Furthermore, from $S\nu_{\infty}^{\frac{1}{3}}\leq\mu_{\infty}$, we have

We claim ($1^*$) holds. In fact, if ($2^*$) holds, it follows from (2.5), (2.9) and $s(x) > 1$ that

From (2.20) and (2.31), we have

Choose $\lambda _{2}$ small enough such that $\frac{1}{4}\mu_{\infty}+\frac{1}{12}\nu_{\infty}-C_\varepsilon\lambda^{\frac{6}{6-p}} > 0$ for all $\lambda\in (0, \lambda_{2})$, which is a contradiction. Thus we have $\mu_{\infty} = \nu_{\infty} = 0$. By the definitions of $\mu_{\infty}$ and $\nu_{\infty}$, we obtain

Hence,

Let $R\rightarrow\infty$ in (2.34), from (2.22) and (2.33), we get

it follows that

On one hand, since $\{u_{n}\}$ is bounded in $D^{1, 2}$, we set $U = \lim\limits_{n\rightarrow\infty}\|u_{n}\|.$ Since $\langle I'(u_{n}), u_{n}\rangle\rightarrow0\; as\; n\rightarrow\infty,$ utilizing (2.9) and (2.35), we have

Then by Lemma 2.1, we have

On the other hand, since $\{u_{n}\}$ is a $(PS)_{c}$ sequence for $I$, i.e., $\langle I'(u_{n}), v\rangle\rightarrow0$ $\, as\; n\rightarrow\infty$ for all$\, v\in D^{1, 2}$, that implies

Combining (2.9), Lemma 2.1 with the fact of $u_{n}\rightharpoonup u\; \text{in}\; D^{1, 2}$, we have

Taking $v = u$ in (2.38), we get

Comparing (2.37) with (2.39), we get $\|u\| = U = \lim\limits_{n\rightarrow\infty}\|u_{n}\|.$ Noticing that $D^{1, 2}$ is a reflexive Banach space, combining above analysis, we can prove that $u_{n}\rightarrow u\; \text{in}\; D^{1, 2}\; as\; n\rightarrow\infty$. Taking $\lambda^{**} = \text{min} \{\lambda_{1}, \lambda_{2}\}$, we conclude that $I(u)$ satisfies the $(PS)_{c}$ condition for all $\lambda\in (0, \lambda^{**}).$

In order to continue our proof, we will introduce a truncated functional. By $(A_{1})$, Sobolev embedding theorem and above analysis, we have

for all $u\in D^{1, 2}.$ Let $g(t) = C_{6}t^{2}-C_{7}t^{4}-\lambda C_{8} t^{p}-C_{9}t^{6}$. Next we will discuss some properties of $g(t)$.

First of all, it is easy to see that there exist positive constants $\lambda_{3}, \; T_{1}\; \text{and}\; T_{2}\; (T_{1} < T_{2})$ such that for any $\lambda\in (0, \lambda_{3})$, $g(t)$ can take positive maximum value for some $t > 0$, and we have

Let $\tau:\mathbb{R}^{+}\rightarrow[0, 1]$ be $C^{\infty}$ function such that

Now, we give the truncated functional as follows:

Since $\tau\in C^{\infty}$, we get $I_{\infty}(u)\in C^{1}(D^{1, 2}, \mathbb{R})$. Similar to above analysis, we obtain

where constants $C_{6}, \cdots, C_{9}$ are the same as those in (2.40).

Let $g_{\infty}(t) = C_{6}t^{2}-C_{7}\tau(t)t^{4}-\lambda C_{8}t^{p}-C_{9}\tau(t)t^{6}$. We say that $g_{\infty}(t)\geq g(t)$ for all $t > 0$. In fact, if $0\leq t\leq T_{1}$, $g_{\infty}(t) = g(t)$; If $T_{1} < t < T_{2}$, $0 < g(t) < g_{\infty}(t)$; If $t\geq T_{2}, \; g_{\infty}(t) > 0\geq g(t)$. Moreover, we obtain that $I_{\infty}(u) = I(u)$ when $0\leq\|u\|\leq T_{1}$.

Lemma 2.4. If $u$ satisfies that $I_{\infty}(u) < 0$, then $\|u\|\leq T_{1}$ and there exists $\epsilon > 0$ such that for all $v\in B_{\epsilon}(u)$, there holds $I_{\infty}(v) = I(v).$ Furthermore, there exists $\lambda^{*} > 0$ such that for all $\lambda\in(0, \lambda^{\ast})$, $I_{\infty}(u)$ satisfies the $(PS)_{c}$ condition for $c < 0$.

Proof. We prove by contradiction. If $I_{\infty}(u) < 0$ and $\|u\| > T_{1}$, from above analysis, we get $I_{\infty}(u)\geq g_{\infty}(\|u\|) > 0$, which is a contradiction, thus we obtain $\|u\|\leq T_{1}$. Since $I(u)$ and $I_{\infty}(u)$ are both continuous, we have $I(v) = I_{\infty}(v)$ for all $\; v\in B_{\epsilon}(u)$. Setting $\lambda^{*} = \text{min}\{\lambda^{**}, \lambda_{3}\}$, by using Lemma 2.3, we get $I_{\infty}(u)$ satisfies the $(PS)_{c}$ condition for $c < 0$.

To prove our main results, we need the following deformation lemma.

Lemma 2.5. (^[20]) Let $Y$ be a Banach space, $f\in C^{1}(Y, \mathbb{R})$, $c\in \mathbb{R}$ and $N$ is any neighborhood of $K_{c}\triangleq\{u\in Y|\; f(u) = c, f'(u) = 0\}$. If $f$ satisfies the $(PS)_{c}$ condition, then there exist $\eta_{t}(u)\equiv\eta(t, u)\in C([0, 1]\times Y, Y)$ and constants $\overline{\epsilon} > \epsilon > 0$ such that

$\rm (1)$ $\eta_{0}(u) = u$, $\forall\; u\in Y$,

$\rm(2)$ $\eta_{t}(u) = u$, $\forall\; u\notin f^{-1}[c-\overline{\epsilon}, c+\overline{\epsilon}]$,

$\rm(3)$ $\eta_{t}(u) = u$ is a homeomorphism of $Y$ onto $Y$, $\forall\; t\in[0, 1]$,

$\rm(4)$ $f(\eta_{t}(u))\leq f(u)$, $\forall\; u\in Y$ and $\forall\; t\in[0, 1]$,

$\rm(5)$ $\eta_{1}(f^{c+\epsilon}\setminus N)\subset f^{c-\epsilon}$, where $f^{c} = \{u\in Y| \; f(u)\leq c\}$, $\forall\; c\in < /italic > < italic > \mathbb{R}$,

$\rm(6)$ if $K_{c} = \emptyset$, $\eta_{1}(f^{c+\epsilon})\subset f^{c-\epsilon}$,

$\rm(7)$ if $f$ is even, $\eta_{t}$ is odd in $u$.

At the end of this section, we point out some concepts and results about $Z_{2}$ index theory. Let $Y$ be a Banach space and set

and

where $\gamma(A)$ is the $Z_{2}$ genus of $A$ defined by

In the following lemma, we give the main properties of genus.

Lemma 2.6. (^[20]) Let $A, B\in\Sigma$.

$\rm (1)$ If there exists an odd map $f\in C(A, B)$, then $\gamma(A)\leq\gamma(B)$.

$\rm(2)$ If $A\subset B$, then $\gamma(A)\leq\gamma(B)$.

$\rm(3)$ If there exists an odd homeomorphism between $A$ and $B$, then $\gamma(A) = \gamma(B)$.

$\rm(4)$ If $S^{N-1}$ is the sphere in $\mathbb{R}^{N}$, then $\gamma(S^{N-1}) = N$.

$\rm(5)$ $\gamma(A\cup B)\leq\gamma(A)+\gamma(B)$.

$\rm(6)$ If $\gamma(A) < \infty$, then $\gamma(\overline{A-B})\geq\gamma(A)-\gamma(B)$.

$\rm(7)$ If $A$ is compact, then $\gamma(A) < \infty$, and there exists $\delta > 0$ such that $\gamma(A) = \gamma(N_{\delta}(A))$, where $N_{\delta}(A) = \{x\in Y|\; dist(x, A)\leq\delta\}$.

$\rm(8)$ If $Y_{0}$ is a subspace of $Y$ with codimension $k$, and $\gamma(A) > k$, then $A\cap Y_{0}\neq\emptyset$.

3.   Proof of Theorem 1.1

Now, we will use the genus argument to prove Theorem 1.1.

For any $k\in \mathbb{N}$, we define

Moreover, by the definition, we get $\Sigma_{k+1}\subset\Sigma_{k}$, so we have $c _{k}\leq c _{k+1}.$

Firstly, we prove for any $k\in \mathbb{N}$, there exists $\varepsilon = \varepsilon(k) > 0$ such that

where $I_{\infty}^{-\varepsilon}(u) = \{u\in D^{1, 2}|\; I_{\infty}(u)\leq-\varepsilon\}.$ Let $\Omega$ be an open bounded subset of $\mathbb{R}^{3}$ with smooth boundary and $\ h(x) > 0\ \text{in}\ \Omega$. For fixed $k\in \mathbb{N},$ let $X_{k}$ be a $k$-dimension subspace of $D^{1, 2}(\Omega)$. Choosing $u\in X_{k}$ with $\|u\| = 1$, for $0 < \rho\leq T_{1}\ (T_{1}$ is the same as before), we get

Since $X_{k}$ is a finite dimension space, all the norms are equivalent. For each $u\in X_{k}$ with $\|u\| = 1$, by $(A_{1})$, we know that there exists $\alpha_{k} > 0$ such that

Define

It is easy to check that $\lim\limits_{k\rightarrow\infty}\beta_k = 0$. Hence, by (3.1)–(3.3), we get

Since $1 < p < 2$, for $\lambda\in(0, \lambda ^{*})$, $u\in X_{k}$ with $\|u\| = 1$, there must be $\rho_{0}\in (0, T_{1})$ small enough such that

and $\varepsilon = -\frac{1}{2}\rho_{0}^{2}+\frac{\lambda }{p}\rho_{0}^{p}\alpha_{k}+\frac{1}{6}\rho_{0}^{6}\beta_{k} > 0$.

Let $K_{c} = \{u\in D^{1, 2}|\; I_{\infty}(u) = c, I'_{\infty}(u) = 0\}$ and $S_{\rho_{0}} = \{u\in D^{1, 2}(\Omega)|\; \|u\| = \rho_{0}\}$, then $S_{\rho_{0}}\cap X_{k}\subset I^{-\varepsilon}_{\infty}$. From Lemma 2.6, we have that

Thus we obtain $I^{-\varepsilon}_{\infty}(u)\subset \Sigma_{k}$ and $c = c_{k}\leq-\varepsilon < 0$. Using Lemma 2.4, we know $I_{\infty}(u)$ satisfies the $(PS)_{c}$ condition if $c < 0$, which implies $K_{c}$ is a compact set.

Next, by using the idea in ^[11,12], we give two claims, which are crucial to prove Theorem 1.1.

Claim 1. If $k, l\in \mathbb{N}$ are such that $c = c_{k} = c_{k+1} = \cdots = c_{k+l},$ then $\gamma(K_{c})\geq l+1.$

Arguing by contradiction that $\gamma(K_{c})\leq l$, then there exists a closed, symmetric set $U$ with $K_{c}\subset U$ and $\gamma(U)\leq l$. Since $I_{\infty}(u)$ is even, by Lemma 2.5, we can assume an odd homeomorphism

such that $\eta(I^{c+\delta}_{\infty}\backslash U)\subset I^{c-\delta}_{\infty}$ for some $\delta\in (0, -c)$. By the hypothesis $c = c_{k+l}$, we know there exists an $A\in \Sigma_{k+l}$ such that

that is to say $A\in I^{c+\delta}_{\infty}$. Furthermore, we get

By Lemma 2.6, we know

Therefore, $\overline{\eta(A\backslash U)}\subset \Sigma_{k}$. Then from (3.5) we can obtain

which is a contradiction. Thus we complete the proof of Claim 1.

Claim 2. If $c_{k} < 0$ is a critical value of $I_{\infty}(u)$, then there exists a subsequence of $\{c_k\}$, still denoted by $\{c_{k}\}$ $(k\in\mathbb{N})$, which satisfies

Indeed, since $I_{\infty}(u)$ is bounded from below, it holds that $c_{k} > -\infty$, and we know that $\Sigma_{k+1}\subset\Sigma_{k}$ and $c_{k}\leq c_{k+1} < 0$. Therefore $\{c_{k}\}$ has a limit, denoted by $c_{\infty}$ and $c_{\infty}\leq0$. If $c_{\infty} < 0$, we set

From above analysis, we know that $K$ is compact, symmetric and $0\notin K$ on account of $c_{\infty} < 0$. By Lemma 2.6 (7), we choose $\delta > 0$ small enough such that

where $N_{\delta}(K) = \{u\in D^{1, 2}|\; dist(u, K)\leq\delta\}$. By Lemma 2.5 (5) with $c = c_{\infty}$, there exist $\varepsilon > 0$ and $\eta_{1}$ such that

Fix an integer $q\in \mathbb{N}$ such that

Choose $\hat{A}\in \Sigma_{m+q}$ such that

Setting $B = \overline{\hat{A}\backslash N_{\delta}(K)}$, using (3.6) and (3.8), we have

It follows from Lemma 2.6 that $\gamma(B)\geq \gamma(\hat{A})-\gamma(N_{\delta}(K))\geq q$, so $B\in \Sigma_{q}$. Denoting $D = \eta_{1}(B)$, then we have $D\in \Sigma_{q}$. Using (3.7) and (3.9), we get

which is absurd. Therefore, $c_{\infty} = 0$.

Now, we conclude the proof of Theorem1.1. For all $k\in \mathbb{N}$, we have $\Sigma_{k+1}\subset\Sigma_{k}$ and $c_{k}\leq c_{k+1} < 0$. If every $c_{k}$ is distinct, then $\gamma(K_{c_{k}})\geq1$ and we know $\{c_{k}\}$ is a sequence of distinct negative critical values of $I_{\infty}(u)$. If for some $k_{0}\in \mathbb{N}$, there exists a $l\geq1$ such that $c = c_{k_{0}} = c_{k_{0}+1} = \cdots = c_{k_{0}+l}$, then by Claim 1, we obtain $\gamma(K_{c})\geq l+1$, which implies that $K_{c}$ contains infinitely many distinct elements. Moreover, by Claim 2, we know there exists a subsequence of $\{c_{k}\}$, still denoted by $\{c_{k}\}$, satisfying $c_{k}\rightarrow0\; \text{as}\; k\rightarrow\infty$. By Lemma 2.4 we know that $I(u) = I_{\infty}(u)$ if $I_{\infty}(u) < 0$. Hence we conclude that there exist infinitely many critical points of $I(u)$ and the sequence of the negative critical values converges to zero. Thus, we complete our proof of Theorem 1.1.

4.   Proof of Theorem 1.2

We denote $D^{1, 2}_{T} = \{u\in D^{1, 2}:u(\tau x) = u(x), \tau\in O(3)\}$ and $L_{T}^{6} = \{u\in L^{6}:u(\tau x) = u(x), \tau\in O(3)\}$, where $T\subset O(3)$ is a subgroup. By the principle of symmetric criticality, we have the following results.

Lemma 4.1. (^[25]) If $I'(u) = 0$ in $D^{1, 2}_{T}$, then $I'(u) = 0$ in $D^{1, 2}$.

Lemma 4.2. If $|T| = \infty, s(0) = 0$ and $\lim\limits_{|x|\rightarrow\infty}s(x) = 0$, then $I$ satisfies the $(PS)_{c}$ condition for all $c\in\mathbb{R}$, where $|T|: = \inf \limits_{x\in \mathbb{R}^3, x\neq0}|T_x|$.

Proof. Since the proof is similar to Lemma 2.3, we just give a sketch of the proof. Let $\{u_n\}$ be a $(PS)_c$ sequence of $I$. An argument similar to the one used in proving Lemma 2.3 shows that $\{u_n\}$ is bounded and there exists a measure $\nu$ such that (2.10) holds. We claim that the concentration of $\nu$ cannot occur at any $a\neq0$ $(a\in\mathbb{R}^3)$. Assuming that $a_k\neq0$ is a singular point of $\nu$, we can obtain $\nu_k = \nu(a_k) > 0$. Since $\nu$ is $T$-invariant, then $\nu(\tau a_k) = \nu_k$ for all $\tau\in T$. And we can know the sum in (2.11) is infinite due to $|T| = \infty$, which is a contradiction. On the other hand, by $\nu_i\leq s(a_i)\nu_i$ and $s(0) = 0$, we get $\nu_0: = \nu(0) = 0$.

Next, we prove that the concentration of $\nu$ cannot occur at infinity. Since $\lim\limits_{|x|\rightarrow\infty} s(x) = 0$, we deduce that

By (2.29), we have $\mu_{\infty} = 0$. From $S\nu_{\infty}^{\frac{1}{3}}\; \leq\; \mu_{\infty}$, we obtain $\nu_\infty = 0$. Thus we get $u_n\rightarrow u$ in $D^{1, 2}_T$ as $n\rightarrow\infty$.

Since $D^{1, 2}_T$ is a separable Banach space, there exists a linearly independent sequence $\{e_j\}$ such that

Denote $Y_{k} = \bigoplus\limits_{j\leq k}D^{1, 2}_j$ and $Z_k = \overline{\bigoplus\limits_{j\geq k}D^{1, 2}_j}.$

Lemma 4.3. (^[25]) Let $I\in C^1(D^{1, 2}_T, \mathbb{R})$ be an even functional satisfying the $(PS)_c$ condition for every $c > 0$. If for every $k\in\mathbb{N}$ there exist $\rho_k > r_k > 0$ such that

$\bf(a)$ $\alpha_k: = \max\limits_{u\in Y_k, \|u\| = \rho_k}I(u)\leq0$,

$\bf(b)$ $\beta_k: = \inf\limits_{u\in Z_k, \|u\| = r_k}I(u)\rightarrow\infty$ as $k\rightarrow\infty$,

then $I$ has a sequence of critical values which converges to $\infty$.

Proof of Theorem 1.2. It is easy to see that $I(u)$ is even and $I(u)\in C^1(D^{1, 2}_T, \mathbb{R})$. By Lemma 4.2, we know $I(u)$ satisfies the $(PS)_c$ condition for every $c > 0$. From the definition of $Y_k$ and $s(x) > 0$ a.e. in $\mathbb{R}^3$, which imply that there exists a constant $\varepsilon_k > 0$ such that for all $w\in Y_k$ with $\|w\| = 1$, we have

On the one hand,

Hence if $u\in Y_k, u\neq0$ and writing $u = t_kw$ with $\|w\| = 1$, by (4.1) and (4.2), we have

for $t_k$ large enough. Thus we have proved (a) of Lemma 4.3.

In the following part, we want to verify (b) of Lemma 4.3. Define

and

It is clear that $0\leq\upsilon_{k+1}\leq\upsilon_k$ and $\upsilon_k\rightarrow\upsilon_0\geq0$. And for every $k\geq1$, there exists a $u_k\in Z_k$ with $\|u_k\| = 1$ such that

By the definition of $Z_k$, we get $u_k\rightharpoonup0$ as $k\rightarrow\infty$ in $D^{1, 2}_T$. Therefore, there exists $\nu$ such that (2.11) holds. Combining the arguments used in Lemma 4.2 with the fact that $|T| = \infty$, we see that the concentration of the measure $\nu$ can only occur at $0$ and $\infty$, thus we have $u_k\rightarrow0$ in $L^6(\Omega)$, where $\Omega = \{x\in\mathbb{R}^3:r < |x| < R\}$ for each $0 < r < R$. Since $s(0) = 0, \lim\limits_{|x|\rightarrow\infty}s(x) = 0$, by $(A_3)$, for each $\varepsilon > 0$, we can choose $r$ small and $R$ large, such that

Hence by Sobolev embedding theorem, we can obtain

Using (4.3), we get $\upsilon_0 = 0$.

By Lemma 2.1(5), we obtain $\gamma _k\rightarrow0\; as\; k\rightarrow\infty$. Since $h(x)\geq0$, $s(x) > 0$ a.e. in $\mathbb{R}^3$ and $\lambda > 0$, for $u\in Z_k$, by (4.3), Sobolev and Young inequalities, we have

On the other hand, since $1 < p < 2$, then there exists $R > 0$ such that $\frac{1}{4}\|u\|^2\geq \frac{1}{192}+\frac{\lambda}{p}\|h(x)\|_{\frac{6-p}{6}} \|u\|^p$ for any $\|u\|\geq R$. Taking $\|u\| = r_k: = \left(\frac{3}{16\gamma_k^6+4\upsilon_k^6}\right)^{\frac{1}{4}},$ by $\upsilon_k\rightarrow0$ and $\gamma_k\rightarrow0$, we get $r_k\rightarrow\infty$ as $k\rightarrow\infty.$ Furthermore, we have

This concludes the proof of Theorem 1.2.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11171220).

Conflict of interest

All authors declare no conflicts of interest in this paper.