Schrödinger-Poisson system without growth and the Ambrosetti-Rabinowitz conditions

1.   Introduction

In this paper, we are interested in finding nontrivial solutions to the following one-parameter family of Schrödinger-Poisson system:

In recent years, system (1.1) has been studied widely due to the fact that it arises in several physical phenomena (see ^[3,4,5,9,18]). From the viewpoint of quantum mechanics, this system describes a charged wave interacting with its own electrostatic field in the case that magnetic effects could be ignored. The terms $u$ and $\phi$ describe the wave functions associated to the particle and electric potential. The term $\phi u$ is nonlocal and concerns the interactions with electric field. The nonlinearity models the interaction between the particles and external nonlinear perturbations.

There has been a lot of contributions about the following Schrödinger-Poisson system

which was first introduced in ^[4]. The case $V(x) = 1$ and $f(x, u) = |u|^{p-2}u$, $2 < p < 6$, has been studied in ^[7], where Ruiz gave existence and nonexistence results. The existence of a ground state solution of (1.2) with $f(x, u) = |u|^{p-2}u$ and $3 < p < 6$ was proved by Azzollini ^[1]. For the general nonlinearity $f$ and the potential $V(x)$, in ^{[6,12,13,15,20,21]}, the authors studied the existence and multiplicity of nontrivial solutions for the Schrödinger-Poisson system with superlinear and subcritical growth condition. The following global Ambrosetti-Rabinowitz type condition plays a crucial role in the above mentioned papers:

where $\gamma > 4$. Since the nonlocal term $\int_{\mathbb{R}^{3}}\phi_{u}u^{2}$ in the energy functional of (1.2) is homogeneous of degree 4, if $\gamma > 4$ from (A-R) then Ambrosetti-Rabinowitz condition guarantees boundedness of Palais-Smale sequences as well as existence of a mountain pass geometry.

It is very natural for us to pose the question: Can we replace (A-R) with a weaker condition? When $V(x)$ is periodic or asymptotically periodic and $f(u)$ does not satisfy the Ambrosetti-Rabinowitz condition, Alves, Souto and Soares ^[2] established the existence of positive ground state solutions by using the mountain pass theorem. In ^[14] Mao et al. studied the following Schrödinger-Poisson system of the form

where $f$ satisfies $0 < 4F(s)\leq sf(s)$, for $s > 0$ is small. Under the conditions that $\varepsilon$ is small and $\lambda$ is large, the authors proved the existence of a positive solution. Differently from the above-mentioned results, the purpose of this paper is to present some existence and multiplicity results of solutions of problem (1.1) under the nonlinearity $f(t)$ which possesses only conditions in a neighborhood of the origin. More importantly, we consider the case that $f$ satisfies $0 < \gamma F(t)\leq tf(t)$ where $\gamma\in(3, 4]$, for $t > 0$ is small. To the best of our knowledge, there are less results in the literatures on the case $\gamma\in(3, 4)$.

Firstly, we study problem (1.1) under the following conditions:

$(V_1)$: $V\in C(\mathbb{R}^{3}, \mathbb{R})$, $0 < V_{L}\leq V(x)$ for all $x\in \mathbb{R}^{3}$ and $V(x)$ is coercive, $i.e.,$ $\lim\limits_{|x|\to\infty}V(x) = \infty$;

$(V_2)$: $V\in C^1(\mathbb{R}^{3}, \mathbb{R})$ and $2V(x)+\nabla V(x)\cdot x\geq0$ for $a.e.$ $x\in \mathbb{R}^{3}$ and $\nabla V(x)\cdot x\in L^{r}(\mathbb{R}^3)$ for some $r\in[\frac{3}{2}, \infty]$.

$(f_0)$: $f(t) = 0$ for $t\leq0$;

$(f_1)$: there exists $\alpha\in(4, 2^\ast)$ such that $\limsup\limits_{t\to0^{+}}\frac{f(t)t}{t^{\alpha}} < +\infty$, where $2^{\ast} = 6$;

$(f_2)$: there exists $\beta\in(4, 2^\ast)$ such that $\liminf\limits_{t\to0^{+}}\frac{F(t)}{t^{\beta}} > 0$;

$(f_3)$: there exists $3 < \gamma\leq4$ such that $0 < \gamma F(t)\leq tf(t), \ \mbox{for}\ t > 0\ \mbox{is small}$;

$(f'_1)$: there exists $\alpha\in(4, 2^\ast)$ such that $\limsup\limits_{t\to0}\frac{f(t)t}{|t|^{\alpha}} < +\infty$;

$(f'_2)$: there exists $\beta\in(4, 2^\ast)$ such that $\liminf\limits_{t\to0}\frac{F(t)}{|t|^{\beta}} > 0$, where $F(t) = \int_{0}^{t}f(s)ds$;

$(f'_{3})$: there exists $\widetilde{\gamma} > 4$ such that $0 < \widetilde{\gamma} F(t)\leq tf(t), \ \mbox{for}\ |t|\ \mbox{small and}\ t\neq0$;

$(f_{4})$: $f(-t) = -f(t)$, for $|t|$ small.

Next, we give our main results.

Theorem 1.1. Assume $(V_1)$, $(V_2)$ and $(f_0)-(f_3)$ hold. Then the problem (1.1) has a positive solution $u\in X$ ($X$ is defined in Sect.2) for all sufficiently large $\lambda$.

Remark 1.1. $(V_2)$ is used to obtain a special bounded Palais-Smale sequence with Jeanjean's monotonicity trick. That $\nabla V(x)\cdot x\in L^{r}(\mathbb{R}^{3})$ for some $r\in[\frac{3}{2}, \infty]$ plays an important role in deriving the Pohozaev identity for the weak solutions of (2.2).

To prove Theorem 1.1, we are faced with several difficulties. On one hand, due to the nonlinearity $f$ without any growth condition at infinity, the natural variational functional associated to (1.1) may be not well defined. Inspired by work of Costa ^[8] and Huang ^[10], we modify $f(t)$ to a new well-defined nonlinearity. Furthermore by Moser iteration, we shall show that for large $\lambda$, the solutions of the modified problem are the solutions of the original problem.

On the other hand, different from ^[8] and ^[14], since we don't assume the global Ambrosetti-Rabinowitz condition about $f(t)$, the boundedness of Palais-Smale sequence seems hard to verify. We use an argument developed by Jeanjean ^[11] to overcome this difficulty. Then Pohozaev type identity ^[19] and the condition $(V_2)$ are used to construct a special bounded Palais-Smale sequence for the modified functional $J_{\lambda}$ (will be defined in Section 2).

Theorem 1.2. Assume $(V_1)$, $(f'_1)-(f'_{3})$ and $(f_{4})$ hold. Then for any given positive integer $k\geq1$ the problem (1.1) has $k$ pairs of solutions $\pm u_i\in X$($i = 1, 2, ..., k$) for all sufficiently large $\lambda$.

The key to prove Theorem 1.2 is a priori estimate of the weak solution for the modified problem. Firstly, we modify $f$ to a new nonlinearity $h$ which is odd and satisfies the Ambrosetti-Rabinowitz condition (see Lemma 2.1). By symmetric mountain pass theorem, the modified problem has a sequence of weak solutions. Secondly, it will be shown that the solutions converge to zero in $L^{\infty}$-norm as $\lambda\to\infty$. Therefore, for $\lambda$ large, they are solutions of the original problem.

Remark 1.2. It is evident that the following function satisfies hypotheses $(f'_1)-(f'_3)$ and $(f_4)$:

where $4 < \alpha < 2^{\ast} < q < \infty$ and $C_{1}, \ C_{2}$ are positive constants.

This paper is organized as follows. In Section 2, we describe the related mathematical tools and give the proof of Theorem 1.1. Theorem 1.2 is proved in Section 3.

In what follows, $C$ and $C_i$ will denote positive generic constants.

2.   Proof of Theorem 1.1

As usual, the norm of $L^{s}(\mathbb{R}^N)$ ($s\geq1$) is denoted by $|\cdot|_{s}$. Define

endowed with the following norm

By $(V_1)$, it well known that $X\hookrightarrow L^{p}(\mathbb{R}^3)$ continuously for $p\in [2,6]$, compactly for $p\in[2, 6)$.

By the conditions $(f_{0})$, $(f_{1})$ and $(f_{2})$, there exist positive constants $\delta\in(0, \frac{1}{2}), \ C_{3}$ and $C_{4}$ such that

For the fixed $\delta > 0$, we now consider $d(t)\in C^{1}(\mathbb{R}, \mathbb{R})$ is a cut-off function satisfying

$|d'(t)|\leq\frac{2}{\delta}$ and $0\leq d(t)\leq1$ for $t\in[\delta, 2\delta]$. Define $G(t) = d(t)F(t)+(1-d(t))F_{\infty}(t),$ where

Set $g(t) = G'(t)$. We observe that the conditions $(f_{0})$–$(f_{3})$ imply some properties of $g(t)$.

Lemma 2.1. (1) $g\in C(\mathbb{R}, \mathbb{R})$, $g(t) = 0$, for all $t\leq0$ and $g(t) = o(1)$ as $t\to 0^{+}$;

(2) $\lim\limits_{t\to+\infty}\frac{g(t)}{t^{3}} = +\infty$;

(3) there exists $C_5 > 0$ such that $g(t)\leq C_5t^{\alpha-1}$, for all $t\geq0$;

(4) for any $T > 0$, there exists a constant $C(T) > 0$ such that $G(t)\geq C(T)t^{\beta}$ for all $t\in[0, T]$;

(5) for all $t > 0$, we have $0 < \gamma G(t)\leq tg(t)$.

By ^[17], for every $u\in H^{1}(\mathbb{R}^3)$, there exists a unique $\phi_u\in D^{1, 2}(\mathbb{R}^3)$ such that

and

It has the following properties:

Lemma 2.2. For any $u\in X\subset H^{1}(\mathbb{R}^3)$, we have

(1) $\phi_u\geq0;$

(2) $\phi_{tu} = t^2\phi_u$;

(3) $\|\phi_u\|^{2}_{D^{1, 2}} = \int_{\mathbb{R}^3}\phi_{u}u^2dx\leq C_6|u|^{4}_{12/5}\leq C_7\|u\|^{4}$, where $C_6$, $C_7$ are constants.

We now consider the modified equation of (1.1) given by

By definition of $G$ and Lemma 2.1, for $u\in X$, the functional associated to (2.2) given by

is well-defined.

Noticing that we can not ensure that the modified nonlinearity $g$ satisfies the classical Ambrosetti-Rabinowitz condition, the boundedness of Palais-Smale sequence seems hard to prove. The following abstract result ^[11] is used to construct a special Palais-Smale sequence.

Proposition 1. Let $X$ be a Banach space equipped with a norm $\|\cdot\|_{X}$ and let $\mathfrak{J}\subset \mathbb{R}^{+}$ be an interval. $\{\Phi_{\mu}\}_{\mu\in\mathfrak{J}}$ are $C^{1}$-functionals on $X$ of the form

where $B(u)\geq0$ for all $u\in X$ and either $A(u)\to+\infty$ or $B(u)\to+\infty$ as $\|u\|_{X}\to+\infty$. Suppose that there exist two points $u_1$, $u_2\in X$ such that

where $\Gamma = \{\gamma\in C([0, 1], X):\ \gamma(0) = u_{1}, \ \gamma(1) = u_{2}\}.$ Then, for almost every $\mu\in\mathfrak{J}$, there exists a sequence $\{u_n(\mu)\}\subset X$ such that

(1) $\{u_n(\mu)\}$ is bounded in $X$,

(2) $\Phi_{\mu}(u_{n}(\mu))\to c_{\mu}$,

(3) $\Phi'_{\mu}(u_{n}(\mu))\to 0$, in $X^{\ast}$, where $X^{\ast}$ is dual space of $X$.

Furthermore, the map $\mu\to c_{\mu}$ is continuous from the left.

Consider a family of functionals

Denote $A(u) = \frac{1}{2}\int_{\mathbb{R}^3}\left(|\nabla u|^{2}+V(x)u^{2}\right)dx+\frac{1}{4}\int_{\mathbb{R}^3}\phi_u(x)u^2dx$, $B(u) = \int_{\mathbb{R}^{3}}\lambda G(u)dx$ and $\mathfrak{J} = [\frac{1}{2}, 1]$. Then $J_{\mu, \lambda}(u) = A(u)-\mu B(u)$. The next lemma ensures that $J_{\mu, \lambda}$ satisfies all assumptions of Proposition 1.

Lemma 2.3. Assume that $(V_{1})$ and $(f_0)-(f_2)$ hold. For all $u\in X$, then

(1) $B(u)\geq0$;

(2) $A(u)\to\infty$ as $\|u\|\to\infty$;

(3) there exists $u_0\in X$, independent of $\mu$, such that $J_{\mu, \lambda}(u_0) < 0$ for all $\mu\in[\frac{1}{2}, 1]$;

(4) for all $\mu\in[\frac{1}{2}, 1]$, it holds

where $\Gamma = \{\gamma\in C([0, 1], X):\ \gamma(0) = 0, \ \gamma(1) = u_{0}\}.$

Proof. From Lemma 2.1-(1) and $(V_1)$, (1) and (2) are proved directly. To prove (3), let us fix some nonnegative radially symmetric function $e(x)\in C^{\infty}_{0}(\mathbb{R}^3)\setminus\{0\}$. Then, for $t > 0$, we have

By Lemma 2.1-(2), it is easy to see that $J_{1/2, \lambda}(te) < 0$ for $t$ large.

It remains to prove (4). By Lemma 2.1-(3) and the Sobolev embedding theorem, we have

From this, we get $c_{\mu} > 0$ and complete the proof.

Remark 2.1. By Lemma 2.3 and Proposition 1, then for almost every $\mu\in[\frac{1}{2}, 1]$, there exists a sequence $\{u_n\}\subset X$ satisfying

Lemma 2.4. The sequence $\{u_n\}$ given in (2.6), up to subsequence, converges to a positive critical point $u_{\mu}$ of $J_{\mu, \lambda}$ with $J_{\mu, \lambda}(u_{\mu}) = c_{\mu}$.

Proof. Since $\{u_n\}$ is bounded in $X$, we have

for some $u_{\mu}\in X$. For all $\varphi\in C_{0}^{\infty}(\mathbb{R}^3)$, using Lebesgue's Theorem, we have that

where we used $u_n\rightharpoonup u_{\mu}$ in $X$, Lemma 2.1-(3) and $u_n\to u_{\mu}$ in $L^{\alpha}(\mathbb{R}^3)$. Thus recalling that $J_{\mu, \lambda}'(u_n)\to0$ we indeed have $J'_{\mu, \lambda}(u_{\mu}) = 0$.

Next we prove $u_n\to u_{\mu}$ in $X$. Using Lemma 2.1-(3) and the fact $u_n\to u_{\mu}$ in $L^{\alpha}(\mathbb{R}^3)$, we get

Hence

where we used the elementary inequalities

Therefore, $u_n\to u_{\mu}$ in $X$. The positivity of $u_{\mu}$ follows by a standard argument (see ^[19]).

To show the above results are true when $\mu = 1$, we need the following remark and lemmas.

Remark 2.2. Assume that $(V_{1})$ and $(f_0)-(f_2)$ hold. Then there exist $\{\mu_n\}\subset[\frac{1}{2}, 1]$ and $\{u_{\mu_n}\}\subset X\setminus\{0\}$ such that $\lim\limits_{n\to+\infty}\mu_{n} = 1$, $u_{\mu_n} > 0$, $J_{\mu_n, \lambda}(u_{\mu_n}) = c_{\mu_{n}}\leq c_{\frac{1}{2}}$ and $J_{\mu_n, \lambda}'(u_{\mu_n}) = 0$.

Lemma 2.5. (See ^[12]) If $u\in X$ is a critical point of $J_{\mu, \lambda}$ and $(V_1)$ holds, then

Lemma 2.6. Assume that $(V_2)$ and $(f_0)-(f_3)$ hold. Then the sequence $\{u_{\mu_{n}}\}$ obtained in Remark 2.2 is bounded with respect to $\mu_n$ in $X$.

Proof. Using the fact $J_{\mu_n, \lambda}(u_{\mu_n})\leq c_{\frac{1}{2}}$, $\langle J'_{\mu_n, \lambda}(u_{\mu_n}), u_{\mu_n}\rangle = 0$ and Lemma 2.5, we have

and

Using $(V_2)$, Lemma 2.1 and the fact that $3 < \gamma\leq 4$, it implies that $\{\int_{\mathbb{R}^3}\phi_{u_{\mu_{n}}}u^{2}_{\mu_{n}}dx\}$ is bounded.

Next we prove that $\|u_{\mu_n}\|$ is bounded. By $\langle J'_{\mu_n, \lambda}(u_{\mu_n}), u_{\mu_n}\rangle = 0$, we obtain

then we complete the proof.

Lemma 2.7. Assume that $(V_1)$, $(V_2)$ and $(f_0)-(f_3)$ hold. Then problem (2.2) has at least one positive solution.

Proof. Using Remark 2.2 and Lemma 2.6, there exist $\{\mu_n\}\subset[\frac{1}{2}, 1]$ and a bounded sequence $\{u_{\mu_{n}}\}\subset X\setminus\{0\}$ such that

Furthermore

where we used the fact that $\mu\mapsto c_{\mu}$ is continuous from the left. By the similar argument, we get

Thus $\{u_{\mu_{n}}\}$ is a bounded Palais-Smale sequence for $J_{\lambda}$ and $\lim\limits_{n\to\infty}J_{\lambda}(u_{\mu_{n}}) = c_1$. By the argument of Lemma 2.4 again, we complete the proof.

Indeed, the critical points of $J_{\lambda}$ with $L^{\infty}$-norm not more than $\delta$ are also the weak solutions of problem (1.1). So next we shall study the $L^{\infty}$-estimates for solutions of (2.2), which is essentially contained in the work of Brezis-Kato.

Lemma 2.8. Assume $(V_{1})$, $(V_{2})$, $(f_0)-(f_3)$ hold and $u\in X$ is a weak solution of problem (2.2). Then $u\in L^{\infty}(\mathbb{R}^3)$. Moreover,

where $C_9 > 0$ only depends on $\alpha$.

Proof. Let $u\in X$ be a weak solution of

which is equivalent to

From the above Lemma 2.7, we know that $u > 0$. Let $T > 0$, and define

Choosing $\varphi = u_{T}^{2(\eta-1)}u$ in (2.13), where $\eta > 1$, we get

Combining Lemma 2.1-(3) and the nonnegativity of the second, the third and the fourth terms in the left side of the above equation, we obtain

On the other hand, by the Sobolev inequality, we obtain

where we used the fact that $(a+b)^{2}\leq 2(a^{2}+b^{2})$.

By (2.14), the Hölder inequality and the Sobolev embedding theorem, we have

where we used the fact that $0\leq u_{T}\leq u$.

In what follows, taking $\zeta = \frac{22^{\ast}}{2^{\ast}-\alpha+2}$, we get

Using the Fatou's lemma, letting $T\to+\infty$, it follows that

Define $\eta_{n+1}\zeta = 2^{\ast}\eta_{n}$, where $n = 0, 1, 2, ...$ and $\eta_{0} = \frac{2^{\ast}+2-\alpha}{2}$. By (2.15) we have

By iteration we have

Thus, we obtain $|u|_{\infty}\leq C_{9}\lambda^{\frac{1}{2^{\ast}-\alpha}}\|u\|^{\frac{2^{\ast}-2}{2^{\ast}-\alpha}}.$

By the similar argument in Lemma 2.6, we can obtian the following lemma.

Lemma 2.9. Let $\lambda > \frac{V_L}{2}$ and $u_{\lambda}$ be a critical point of $J_{\lambda}$ with $J_{\lambda}(u_{\lambda}) = c_{1}$. Then there exists $C_{16} > 0$ (independent of $\lambda$) such that

Proof of Theorem 1.1. Let $u_0\in C^{\infty}_{0}(\mathbb{R}^N)\cap X\setminus\{0\}$ be a nonnegative function such that $J_{\lambda}(u_0) < 0$. Then, the functional $J_{\lambda}$ has the mountain pass geometry and we can define

where $\Gamma = \{\gamma\in C([0, 1], X):\ \gamma(0) = 0, \ \gamma(1) = u_0\}.$ By Lemma 2.7, there exists a positive critical point $u_{\lambda}$ of $J_{\lambda}$ with $J_{\lambda}(u_{\lambda}) = d_{\lambda}$. And more remarkable $d_{\lambda} = c_1$.

Furthermore, taking $T = |u_{0}|_{\infty}$, from Lemma 2.1-(4), we obtain

By (2.11), (2.16) and (2.17), we see that

Then there exists $\lambda_{0} > 0$ such that for all $\lambda > \lambda_{0}$, we get

where $\delta$ is fixed in (2.1). Thus, the $u_{\lambda}$ is a positive solution of the original problem (1.1).

3.   Proof of Theorem 1.2

We start by finding that the conditions $(f'_{1})$ and $(f'_{2})$ imply the existence of positive constants $C_{20}, C_{21}$ such that

and

with $|t|$ small. Consider $\rho(t)\in C^{1}(\mathbb{R}, \mathbb{R})$ an even cut-off function satisfying:

$0\leq \rho(t)\leq1$, $t\rho'(t)\leq0$ and $|t\rho'(t)|\leq\frac{2}{\delta}$, where $0 < \delta < \frac{1}{2}$ is chosen such that (3.1), (3.2) hold for $|t|\leq2\delta$ and $(f_4)$ holds for $|t|\leq\delta$. Define

where $\widetilde{F}_{\infty}(t) = C_{22}|t|^{\alpha}$. From $(f'_{1})$ and the definitions of $\rho(t)$ and $h(t)$, for $u\in X$ we have

Lemma 3.1. (See ^[8]) If $f$ satisfies $(f'_{1})-(f'_{3})$, then for all $t\neq0$, we have

where $\theta = \min\{\alpha, \widetilde{\gamma}\}$.

We now consider another modified equation of (1.1) given by

By the definition of $H(t)$ and (3.3), the functional associated to (3.4) stated by

is well-defined. It is well known that its critical points are the weak solutions of (3.4).

The goal of this section is to prove Theorem 1.2. To this end, we use the Lemma 3.1 to get the boundedness of Palais-Smale sequence. Moreover, by the similar argument in Lemma 2.4, it is easy to show that $I_{\lambda}$ satisfies Palais-Smale condition. These are standard results which can be found in textbooks and no proof is given here.

Lemma 3.2. Assume that $(f'_1)-(f'_3)$ are satisfied. If $u\in X$ is a critical point of $I_{\lambda}$, then

where $C_{24}$ depends on $\theta$.

The proof of the above result is quite similar to the one used in Lemma 2.6 and so is omitted.

Since $X$ is a real, reflexive, and separable Banach space, there exists $\{e_j\}_{j\in \mathbb{N}}\subset X$ such that

We denote

Lemma 3.3. Set $\theta_{k, \lambda} = \sup\limits_{u\in Y_{k}}I_{\lambda}(u)$. If $\lambda > 1$, then

where $C_{25}$ depends on $\alpha$, $\beta$ and $k$.

Proof. We notice that $\delta > 0$ was chosen in Section 3 such that

For $u\in Y_k$, denote $\Omega_1 = \{x\in \mathbb{R}^3:\ |u(x)|\geq2\delta\}$, $\Omega_2 = \{x\in \mathbb{R}^3:\ |u(x)| < 2\delta\}$, and let $u_1 = u|_{\Omega_1}$, $u_2 = u|_{\Omega_2}$. Since all norms in $Y_k$ are equivalent, from (3.8) and (3.9) we obtain

and

where $\widehat{C}(k)$ is a positive constant. By the same reason, we can define

Then for $u\in Y_k$, it follows that

For $\lambda > 1$, by $4 < \alpha\leq\beta < 2^{\ast}$, we have

where $C_{25}: = \overline{C}_{1}(k)+\overline{C}_2(k)$.

Lemma 3.4. Assume that $u\in X$ is a weak solution of problem (3.4). Then $u\in L^{\infty}(\mathbb{R}^N)$. Moreover,

where $C_{27} > 0$ only depends on $\alpha$.

The proof of Lemma 3.4 is quite similar to Lemma 2.8 and so is omitted.

To prove Theorem 1.2, we will apply the following symmetric mountain pass theorem due to Rabinowitz ^[16].

Proposition 2. Let $X$ be an infinite dimensional Banach space, $J\in C^{1}(X, \mathbb{R})$ be even, satisfy (PS) condition and $J(0) = 0$. If $X = Y\bigoplus Z$ with $\dim Y < +\infty$, and $J$ satisfies

(1) there are constants $\rho, \alpha > 0$ such that $J|_{\partial B_{\rho}\cap Z}\geq\alpha$,

(2) for any finite dimensional subspace $W\subset X$, there is an $R = R(W)$ such that $J\leq0$ on $W\setminus B_{R(W)}$, then $J$ has a sequence of critical values.

Remark 3.1. We point out that $I_{\lambda}$ satisfies all assumptions of Proposition 2. By Proposition 2, then ${I_{\lambda}}$ possesses a sequence of critical points.

Proof of Theorem 1.2. Fix an integer $k$. Choose $R > 0$ such that $I_{\lambda}(u)\leq0$ for all $u\in Y_{k}$ with $\|u\|\geq R$, and for all $\lambda\geq1$. For $B_{R} = \{u\in X:\ \|u\| < R\}$, let $D = B_{R}\cap Y_{k}$. Define

Let $i(A)$ be the genus of symmetric subset $A$. For $j\leq k$, we denote

and

From Proposition 2 and Remark 3.1, and under our conditions on $h$, we get

Moreover, they are also critical values of $I_{\lambda}$ and there exist at least $2k$ critical points $\{\pm u_{j, \lambda}\}_{j = 1}^k$ at these critical values. Since $Id\in \Gamma$, the definition of $\Theta_{j}$ and Lemma 3.3, we obtain

By Lemmas 3.2–3.4, we have

Since $4 < \beta < 2^{\ast}$, there exists $\lambda_{1} > 0$ such that $|u_{j, \lambda}|_{\infty}\leq \delta$, for all $\lambda > \lambda_{1}$. Thus, $\pm u_{j, \lambda}(j = 1, 2, \cdots, k)$ are weak solutions of problem (1.1).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11171220). The authors wish to thank the referees and the editor for their valuable comments and suggestions.

Conflict of interest

All authors declare no conflicts of interest in this paper.