Existence of solutions for modified Kirchhoff-type equation without the Ambrosetti-Rabinowitz condition

1.   Introduction

In the first part of this paper, we are dedicated to studying the following modified Kirchhoff-type problem with general nonlinearity:

where $a > 0, b \geq 0$ are two constants and $V: {\mathbb{R}}^{3}\rightarrow{\mathbb{R}}$ is a potential function satisfying:

$(V)$: $V(x)\in C(\mathbb{R}^{3})$, $V_{0}: = \inf\limits_{x\in\mathbb{R}^{3}}V(x) > 0$. Furthermore, for any $M > 0$, there is $r > 0$ such that $B_{r}(y)$ centered at $y$ with radius $r$ satisfying

In addition, we suppose that the function $f(x, t)$ verifies:

$(f_1)$: $f\in C(\mathbb{R}^{3}\times\mathbb{R}, \mathbb{R})$, $|f(x, t)|\leq C_{1}\left(1+|t|^{p-1}\right)$ for some $C_{1} > 0$ and $p\in(4, 12)$;

$(f_2)$: $f(x, t) = o(t)$ uniformly in $x$ as $t\to0$;

$(f_3)$: $F(x, t)/t^{4}\to\infty$ uniformly in $x$ as $|t|\to\infty$, where $F(x, t) = \int_{0}^{t}f(x, s)ds$;

$(f_4)$: $t\to f(x, t)/t^{3}$ is positive for $t\neq0$, strictly decreasing on $(-\infty, 0)$ and strictly increasing on $(0, \infty)$.

Clearly, $(f_{1})$ and $(f_{2})$ show that for any $\varepsilon > 0$, there exists $C_{\varepsilon} > 0$ such that

And $(f_2)$ and $(f_4)$ tell that

which is weaker than the following Ambrosetti-Rabinowitz type condition:

As is well known, the (A–R) condition is very useful in verifying the Palais-Smale condition for the energy functional associated problem (1.1). This is very much crucial in the applications of critical point theory. However, although (A–R) is a quite natural condition, it is somewhat restrictive and eliminates many nonlinearities. For example, the function

does not satisfy (A–R) condition for any $\gamma > 4$. But it satisfies our conditions $(f_1)-(f_4)$. For this reason, there have been efforts to remove (A–R) condition. For an overview of the relevant literature in this direction, we refer to the pioneering papers ^{[1,2,3,4,5,6]}.

Problem (1.1) is a nonlocal problem due to the presence of the term $\int_{{\mathbb{R}}^3}|\nabla u|^2 dx$, and this fact indicates that (1.1) is not a pointwise identity. Moreover, problem (1.1) involves the quasilinear term $u\Delta(u^2)$, whose natural energy functional is not well defined in $H^{1}(\mathbb{R}^{3})\cap D^{1, 2}(\mathbb{R}^{3})$ and variational methods cannot be used directly. These cause some mathematical difficulties, and in the meantime make the study of such a problem more interesting.

Some interesting results by variational methods can be found in ^[7,8,9] for Kirchhoff-type problem. Especially, in recent paper ^[10], Li and Ye studied the following problem:

where $p\in(3, 6)$. And they proved problem (K) has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma.

Thereafter, Guo ^[11] generalized the result in ^[10] to the following Kirchhoff-type problem with general nonlinearity

Guo proved problem (K1) also has a positive ground state solution by using the similar way. But applying Guo's result to problem (K), the condition $3 < p < 6$ in ^[10] can be weakened to $2 < p < 6$.

And two years later, Tang and Chen in ^[12] have obtained a ground state solution of Nehari-Pohozaev type for problem (K1) by using a more direct approach than ^[10,11]. Moreover, Tang and Chen in ^[12] found that it does not seems to be sufficient to prove the inequality $c_{\lambda} < m_{\lambda}$ for $\lambda\in[\delta, 1]$ in Lemma 3.3 of ^[11]. Then by referring to ^[12], we correct this problem in the following Lemma 5.11 of the present paper.

In more recent paper ^[13], under more general assumptions on $V(x)$ than ^[10,11,12], He, Qin and Tang have proved the existence of ground state solutions for problem (K1) by using variational method and some new analytical techniques. Moreover, under general assumptions on the nonlinearity $f(u)$, He, Qin and Wu in ^[14] have obtained the existence of positive solution for problem (K1) by using property of the Pohozaev identity and some delicate analysis.

When $a = 1$ and $b = 0, \; (1.1)$ is reduced to the well known modified nonlinear Schrödinger equation

Solutions of equation (1.5) are standing waves of the following quasilinear Schrödinger equation of the form:

where $V(x)$ is a given potential, $k$ is a real constant, $\alpha$ and $g$ are real functions. The quasilinear Schrödinger Eq (1.6) is derived as models of several physical phenomena, such as ^[15,16,17]. In ^[18], Poppenberg firstly began with the studies for Eq (1.6) in mathematics. For Eq (1.5), there are several common ways to prove existence results, such as, the existence of a positive ground state solution has been studied in ^[19,20] by using a constrained minimization argument; the problem is transformed to a semilinear one in ^[21,22] by a change of variables (dual approach); Nehari method is used to get the existence results of ground state solutions in ^[23]. Especially, in ^[24], the following problem:

was studied via a perturbation method, where $\Omega \subset \mathbb{R}^{N}$ is a bounded smooth domain.

Very recently, Huang and Jia in ^[25] studied the following autonomous modified Kirchhoff-type equation:

where $b\geq 0$, $p > 1$. For $p\in(1, 2]\cup[12, \infty)$, depending on the deduction of some suitable Pohozaev identity, they obtained the nonexistence result for Eq (1.7). And for $p\in(3, 4]$, they proved that the existence of ground state solution for Eq (1.7) by using the Nehari-Pohozaev manifold. But for $p\in(2, 3]$, they didn't give the existence of ground state solution for Eq (1.7). We refine the result in this paper.

We point out that $f(x, t)$ is $C^1$ with respect to $t$ and $f(x, t)$ satisfies the Ambrosetti-Rabinowitz condition are very crucial in some related literatures. Since $f(x, t)$ is not assumed to be differentiable in $t$, the Nehari manifold of the corresponding Euler-Lagrange functional is not a $C^{1}$ functional. And if $f(x, t)$ dose not satisfy the Ambrosetti-Rabinowitz condition, the boundedness of Palais-Smale sequence (or minimizing sequence) seems hard to prove. In this case, their arguments become invalid. The first part of this paper intends to deal with the existence of non-trivial solution to problem (1.1) by the perturbation methods when $f(x, t)$ is $C^1$ in $t$ and (A–R) condition are not established.

Now, we give our first main theorem as follows:

Theorem 1.1. If $(V)$ and $(f_1)-(f_4)$ hold, then problem (1.1) has a nontrivial solution.

Remark 1.1. The condition $(V)$ was firstly introduced by Bartsch and Wang ^[26] to guarantee the compactness of embeddings of the work space. The condition $(V)$ can be replaced by one of the following conditions:

$(V_1)$: $V(x)\in C(\mathbb{R}^{3})$, meas$\{x\in\mathbb{R}^{3}:V(x)\leq M\} < \infty$ for any $M > 0$;

$(V_2)$: $V(x)\in C(\mathbb{R}^{3})$, $V(x)$ is coercive, i.e., $\lim\limits_{|x|\to\infty}V(x) = \infty$.

Remark 1.2. Even though the condition $(V)$ is critical to the proof of the compactness of the minimizing sequence for the energy functional, the existence result can also be obtained when $V$ is a periodic potential because of the concentration-compactness principle.

Suppose that problem (1.1) has a periodic potential $V$ and $V$ satisfies

$(V^{'})$: $V(x)\in C(\mathbb{R}^{3})$ is 1-periodic in $x_{i}$ for $1\leq i\leq 3$, $V_{0}: = \inf\limits_{x\in\mathbb{R}^{3}}V(x) > 0$,

and $f(x, t)$ satisfies

$(f'_1)$: $f(x, t)\in C(\mathbb{R}^{3}\times\mathbb{R}, \mathbb{R})$, $f(x, t)$ is 1-periodic in $x_i$ for $i = 1, 2, 3$ and $|f(x, t)|\leq C_{2}\left(1+|t|^{p-1}\right)$ for some $C_{2} > 0$ and $p\in(4, 12)$.

Our second main result is

Theorem 1.2. Suppose $(V^{'})$, $(f'_1)$ and $(f_2)-(f_4)$ hold. Then equation (1.1) has a nontrivial solution.

In the last part of our paper, we are absorbed in the following modified Kirchhoff-type equations with general nonlinearity:

where $a > 0, b\geq0$, $\widetilde E$ is defined at the beginning of Section 5 and $V(x)$ satisfies:

$(V^{*}_{1})$: $V \in C^{1}(\mathbb{R}^{3}, \mathbb{R})$ and there exists a positive constant $A < a$ such that

where$(\cdot, \cdot)$ is the usual inner product in ${\mathbb{R}}^{3}$;

$(V^{*}_{2})$: there exists a positive constant $V_{\infty}$ such that for all $x \in \mathbb{R}^{3}$,

Moreover, we assume that the function $f(s) \in C^{1}\left(\mathbb{R}^{+}, \mathbb{R}\right)$ verifies:

$(f^{*}_{1})$: $f(s) = o(s)$ as $s \rightarrow 0^{+}$;

$(f^{*}_{2})$: $\lim\limits _{s \rightarrow+\infty} \frac{f^{\prime}(s)}{s^{10}} = 0$;

$(f^{*}_{3})$: $\lim\limits _{s \rightarrow+\infty} \frac{f(s)}{s} = +\infty$;

$(f^{*}_{4})$: $\frac{f(s)}{s}$ is strictly increasing in $(0, +\infty)$.

Since we are only interested in positive solutions, we define $f(s) \equiv 0$ for $s \leq 0$.

Remark 1.3. There are a number of functions which satisfy $(V^{*}_1)-(V^{*}_2)$. For example, $V(x) = V_{\infty}-\frac{A}{8(1+|x|^{2})}$, where $0 < A < \min\{2a, 8V_{\infty}\}$ is a constant. Moreover, by Lemma 5.1 mentioned later, we know that $|f(s)|\leq \varepsilon(|s|+|s|^{11})+C_{\varepsilon}|s|^{p-1}$ for every $\varepsilon > 0$ and $p\in (2, 12)$.

The last main result is given below:

Theorem 1.3. If $(V^{*}_1)-(V^{*}_2)$ and $(f^{*}_{1})-(f^{*}_{4})$ hold, then problem (1.8) has a positive ground state solution.

In order to prove Theorem 1.3, we need to overcome several difficulties. First, since the Ambrosetti-Rabinowitz condition or 4-superlinearity does not hold, for $2 < p < 12$, it is difficult to get the boundedness of any $(PS)$ sequence even if a $(PS)$ sequence has been obtained. To overcome this difficulty, inspired by ^[27,28], we use an indirect approach developed by Jeanjean. Second, the usual Nehari manifold is not suitable because it is difficult to prove the boundedness of the minimizing sequence. So we follow ^[29] to take the minimum on a new manifold, which is obtained by combining the Nehari manifold and the corresponding Pohozaev type identity. Third, since the Sobolev embedding $H_V^{1}({\mathbb{R}}^{3})\hookrightarrow L^{q}({\mathbb{R}}^{3})$ for $q\in[2, 2^{*})$ is not compact, it seems to be hard to get a critical point of the corresponding functional from the bounded $(PS)$ sequence. To solve this difficulty, we need to establish a version of global compactness lemma ^[10].

Remark 1.4. In Theorem 1.3, we especially give the existence result for the case where $p\in(2, 3]$, which has few results for this modified Kirchhoff problems and can be viewed as a partial extension of a main result in ^[10,30], which dealt with the cases of $p\in(3, 6)$ and $p\in(4, 2\times2^{\ast})$, respectively.

This paper is organized as follows. In Section 2, we describe the related mathematical tools. Theorem 1.1 and Theorem 1.2 are proved in Section 3 and in Section 4, respectively. In Section 5 we give the proof of Theorem 1.3.

In the whole paper, $C_i$, $C_{\varepsilon}$ and $C_{\varepsilon}^{\prime}$ always express distinct constants.

2.   Preliminaries

Let $L^{p}(\mathbb{R}^3)$ be the usual Lebesgue space with the norm $\|u\|_{p} = \left(\int_{\mathbb{R}^3}|u|^{p}dx\right)^{\frac{1}{p}}$. And $H^{1}(\mathbb{R}^{3})$ is the completion of $C^{\infty}_{0}(\mathbb{R}^{3})$ with respect to the norm $\|u\|_{H} = \left(\int_{\mathbb{R}^{3}}\left(|\nabla u|^{2}+u^{2}\right)dx\right)^{1/2}$. Moreover, $D^{1, 2}(\mathbb{R}^3)$ is the completion of $C_{0}^{\infty}(\mathbb{R}^3)$ with the norm $\|u\|_{D^{1, 2}} = \left(\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx\right)^{1/2}$.

In order to deal with the perturbation functional $I_{\lambda}$ (see Eq (2.3)), the work space $E$ is defined by

where

endowed with the norm

and $W^{1, 4}(\mathbb{R}^3)$ endowed with the norm

Moreover, when $V(x)\equiv1$, we define

The norm of $E$ is denoted by

Notice that the embedding from $H^{1}_{V}(\mathbb{R}^3)$ into $L^{2}(\mathbb{R}^3)$ is compact (^[26]). Thus, by applying the interpolation inequality, we get that the embedding from $E$ into $L^{s}(\mathbb{R}^{3})$ for $2\leq s < 12$ is compact.

A function $u\in E$ is called a weak solution of problem (1.1), if for all $\varphi\in E$, there holds

which is formally associated to the energy functional given by

for $u\in E$, where $F(x, u) = \int_{0}^{u}f(x, s)ds$.

Remind that $\int_{\mathbb{R}^{3}}u^{2}|\nabla u|^{2}dx$ is not convex and well-defined in $H^{1}_{V}(\mathbb{R}^{3})$, we need to take a perturbation functional of (2.2) given by

From condition $(V)$, (1.3) and (1.4), it is normal to verify that $I_{\lambda}\in C^{1}(E, \mathbb{R})$ and

3.   Proof of Theorem 1.1

First of all, let us briefly describe the proof of Theorem 1.1. We first discuss the properties of the perturbed family of functionals $I_{\lambda}$ on the Nehari manifold

Then we prove that $I_{\lambda}(u_{\lambda}) = \inf_{\mathcal{N}_{\lambda}}I_{\lambda}\ \mbox{is achieved.}$ Moreover, since the Nehari manifold $\mathcal{N}_{\lambda}$ is not a $C^{1}$-manifold, we use the general Nehari theory in ^[31] to prove that the minimizer $u_{\lambda}$ is a critical point of $I_{\lambda}$. Finally, solutions of problem (1.1) can be obtained as limits of critical points of $I_{\lambda}$.

Lemma 3.1. Assume $(V)$ and $(f_1)-(f_4)$ hold. Then, for $\lambda\in(0, 1]$, we get the following results:

(1) For $u\in E\setminus \{0\}$, there exists a unique $t_{u} = t(u) > 0$ such that $m(u): = t_{u}u\in\mathcal{N}_{\lambda}$ and

(2) For all $u\in\mathcal{N}_{\lambda}$, there exists $\alpha_{0} > 0$ such that $\|u\|_{W}\geq\alpha_{0}$;

(3) There exists $\rho > 0$ such that $c: = \inf\limits_{\mathcal{N}_{\lambda}}I_{\lambda}\geq\inf\limits_{S_{\rho}}I_{\lambda} > 0$, where $S_{\rho}: = \{u\in E:\|u\| = \rho\}$;

(4) If $\mathcal{V}\subset E\setminus\{0\}$ is a compact subset, there exists $R > 0$ such that $I_{\lambda}\leq0$ on $\mathcal{W}\setminus B_{R}(0)$, where $\mathcal{W} = \{{\mathbb{R}}^{+} u:u\in\mathcal{V}\}$.

Proof. (1) For any $u\in E\setminus\{0\}$, we define a function $h_{u}(t) = I_{\lambda}(tu)$ for $t\in(0, \infty)$, i.e.,

And since the Sobolev embedding $E\hookrightarrow L^{s}(\mathbb{R}^{3})$ for $s\in[2, 12]$ is continuous, combined with (1.3), for $t > 0$ and small $\varepsilon > 0$ one has

where the constant $C_{3}$ is independent of $t$. Since $u\neq0$ and $p > 4$, then for $t > 0$ small enough, we deduce $h_{u}(t) > 0$.

On the other hand, noticing that $|tu(x)|\to\infty$ if $u(x)\neq0$ and $t\to\infty$, by $(f_3)$ and Fatou's lemma, we get

Hence, $h_{u}(t)$ has a positive maximum and there exists a $t_{u} = t(u) > 0$ such that $h_{u}'(t_{u}) = 0$ and $t_{u}u\in\mathcal{N}_{\lambda}$.

Next, we prove the uniqueness of $t_u$. To this aim, we may suppose that there exists $t_{u}^{\ast} > 0$ with $t_{u}^{\ast}\neq t_{u}$ such that $h_{u}'(t_{u}^{\ast}) = 0$. Then we obtain

This together with

implies that

which contradicts with $(f_4)$.

(2) By $u\in\mathcal{N}_{\lambda}$ and (1.3), for $\varepsilon > 0$ small enough, one has

which implies that there exists a constant $\alpha_{0} > 0$ such that $\|u\|_{W}\geq\alpha_{0} > 0$ for all $u\in\mathcal{N}_{\lambda}$.

(3) For some $\rho > 0$ and $u\in E\setminus \{0\}$ with $\|u\|\leq\rho$, there exists $C > 0$ such that

By $(V)$, $(f_1)$, $(f_2)$ and the Sobolev inequality, without loss of generality, we take $\rho < 1$ small enough and $\varepsilon = \frac{V_0}{4}\min\{a, 1\}$, then

whenever $\|u\|\leq\rho$. For any $u\in\mathcal{N}_{\lambda}$, Lemma 3.1-(1) implies that

Take $s > 0$ with $su\in S_{\rho}$. It follows from (3.2) and (3.3) that

Therefore

(4) Arguing by contradiction, then there must exist $u_n\in\mathcal{V}$ and $v_n = t_{n}u_n$ such that $I_{\lambda}(v_n)\geq0$ for all $n$ and $t_n\to\infty$ as $n\to\infty$. Without loss of generality, we may assume that $\|u_n\| = 1$ for every $u_{n}\in\mathcal{V}$. Up to a subsequence, there exists $u\in E$ with $\|u\| = 1$ such that $u_n\to u$ strongly in $E$. Since $|v_{n}(x)|\to\infty$ if $u(x)\neq0$, by $(f_3)$ and Fatou's lemma, then

which implies that

This is a contradiction.

Now we are ready to study the minimizing sequence for $I_{\lambda}$ on $\mathcal{N}_{\lambda}$.

Lemma 3.2. For fixed $\lambda\in(0, 1]$, let $\{u_n\}\subset\mathcal{N}_{\lambda}$ be a minimizing sequence for $I_{\lambda}$. Then $\{u_n\}$ is bounded in $E$. Moreover, passing to a subsequence there exists $u\in E (u\neq0)$ such that $u_{n}\to u$ in $E$.

Proof. Let $\{u_n\}\subset\mathcal{N}_{\lambda}$ be a minimizing sequence of $I_{\lambda}$, i.e.,

From (3.4), we have

Thus, we deduce $\{\|u_n\|_{H_{V}}\}$ is bounded.

Next, we need to prove that $\{\|u_n\|_{W}\}$ is also bounded. By contradiction, if $\{u_n\}$ is unbounded in $W^{1, 4}(\mathbb{R}^3)$, setting $\omega_n = \|u_n\|_{W}^{-1}u_n$, we have

The proof is divided into two cases as follows:

Case 1: $\omega = 0$. From Lemma 3.1-(1), we see

For any $m > 0$ and setting $v_n = (8m)^{1/4}\omega_{n}$, since $v_{n}\to0$ strongly in $L^{p}(\mathbb{R}^{3})$, we deduce from (1.3) that

So for $n$ large enough, $(8m)^{1/4}\|u_n\|_{W}^{-1}\in(0, 1)$, and

That is, for fixed $\lambda > 0$, from the arbitrariness of $m$, we get $I_{\lambda}(u_n)\to\infty$. This contradicts with $I_{\lambda}(u_n)\to c > 0$.

Case 2: $\omega\neq0$. Due to $\omega \neq0$, the set $\Theta = \{x\in\mathbb{R}^3: \omega(x)\neq0\}$ has a positive Lebesgue measure. For $x\in\Theta$, we have $|u_{n}(x)|\to\infty$. This together with condition $(f_3)$, implies

It follows from $I_{\lambda}(u_n)\to c$, $(f_3)$, Sobolev inequality and Fatou's Lemma that

where $C_{9}$ is a constant independent on $n$. This is impossible.

In both cases, we all get a contradiction. Therefore, $\{u_n\}$ is bounded in $W^{1, 4}(\mathbb{R}^3)$. It follows that $\{u_n\}$ is bounded in $E$, so $u_n\rightharpoonup u$ weakly in $E$ after passing to a subsequence. If $u = 0$, for $n$ large enough and $u_n\in\mathcal{N}_{\lambda}$, we see as in (3.5) that

for all $s > 0$, where $C_{10} = \frac{\lambda}{4}\left(\inf\limits_{u\in\mathcal{N}_{\lambda}}\|u\|_{W}\right)^{4} > 0$. It is a contradiction. Hence $u\neq0$.

Since the embedding $H^{1}_V(\mathbb{R}^3)\hookrightarrow L^{p}(\mathbb{R}^3)$ is compact for each $p\in[2, 12)$, similar to Lemma 2.2 in ^[30], it is well known that $u_{n}\to u$ strongly in $E$.

Lemma 3.3. For fixed $\lambda\in(0, 1]$, there exists $u\in\mathcal{N}_{\lambda}$ such that $I_{\lambda}(u) = \inf\limits_{\mathcal{N}_{\lambda}}I_{\lambda}$.

Proof. Let $\{u_n\}\subset\mathcal{N}_{\lambda}$ be a minimizing sequence of $I_{\lambda}$, then $\{u_n\}$ is bounded in $E$ by lemma {3.2}. Thus, up to a subsequence there exists $u\in E (u\neq0)$ such that $u_n\rightharpoonup u$ in $E$ and $I_{\lambda}'(u) = 0$. It follows that $u\in\mathcal{N}_{\lambda}$. Thus, $I_{\lambda}(u)\geq c > 0$. In order to complete the proof, it suffices to show that $I_{\lambda}(u)\leq c$. Indeed, from (1.4), Fatou's lemma and the weakly lower semi-continuity of norm, we have

The proof is completed.

Let $S$ be the unit sphere in $E$. Define a mapping $m(\omega):S\to\mathcal{N}_{\lambda}$ and a functional $J_{\lambda}(\omega):S\to\mathbb{R}$ by

where $t_\omega$ is as shown in Lemma 3.1-(1). As Proposition 2.9 and Corollary 2.10 in ^[31], the following proposition is a consequence of Lemma 3.1 and the above observation.

Proposition 3.1. Assume $(V)$ and $(f_1)-(f_4)$ hold. For fixed $\lambda\in(0, 1]$, then

(1) $J_{\lambda}\in C^{1}(S, \mathbb{R})$, and

for any $z\in T_{\omega}S = \{v\in E:\langle v, \omega\rangle = 0, \forall \omega\in S\}$;

(2) $\{\omega_{n}\}$ is a Palais-Smale sequence for $J_{\lambda}$ if and only if $\{m(\omega_{n})\}$ is a Palais-Smale sequence for $I_{\lambda}$;

(3) $\omega\in S$ is a critical point of $J_{\lambda}$ if and only if $m(\omega)\in\mathcal{N}$ is a critical point of $I_{\lambda}$. Moreover, the corresponding critical values of $J_{\lambda}$, $I_{\lambda}$ coincide and $c = \inf\limits_{S}J_{\lambda} = \inf\limits_{\mathcal{N}_{\lambda}}I_{\lambda}$.

Finally, for the proof of Theorem 1.1, we need to introduce the following result.

Lemma 3.4. Assume the conditions $(V)$ and $\left(f_{1}\right)-\left(f_{4}\right)$ hold. Let $\left\{\lambda_{n}\right\} \subset(0, 1]$ be such that $\lambda_{n} \rightarrow 0$. Let $\{u_{n}\} \subset E$ be a sequence of critical points of $I_{\lambda_{n}}$ with $I_{\lambda_{n}}\left(u_{n}\right) \leq C$ for some constant $C$ independent of $n$. Then, passing to a subsequence, we have $u_{n} \rightarrow \widetilde{u}$ in $H_{V}^{1}({\mathbb{R}}^{3}), u_{n} \nabla u_{n} \rightarrow \widetilde{u} \nabla \widetilde{u}$ in $L^{2}({\mathbb{R}}^{3}), \lambda_{n} \int_{\mathbb{R}^{3}}\left(\left|\nabla u_{n}\right|^{4}+\right. \; \left.u_{n}^{4}\right) d x \rightarrow 0$, $I_{\lambda_{n}}\left(u_{n}\right) \rightarrow I(\widetilde{u})$ and $\widetilde{u}$ is a critical point of $I$.

Proof. First, similar to Lemma 3.2, we can get $\{u_n\}$ is bounded in $E$. Then, this together with Theorem 3.1 in ^[30] can complete the proof.

Proof of Theorem 1.1 Let $\{\omega_n\}\subset S$ be a minimizing sequence for $J_{\lambda}$. As is mentioned above, we may assume $J_{\lambda}'(\omega_{n})\to0$ and $J_{\lambda}(\omega_n)\to c$ by Ekeland's variational principle. From Proposition 3.1-(2), for $u_n = m(\omega_n)$ we have $I_{\lambda}(u_n)\to c$ and $I_{\lambda}'(u_n)\to0$. Therefore, $\{u_n\}$ is a minimizing sequence for $I_{\lambda}$ on $\mathcal{N}_{\lambda}$ and from Lemma 3.3 there exists a minimizer $u$ of $I_{\lambda}|_{\mathcal{N}_{\lambda}}$. Then $m^{-1}(u)\in S$ is a minimizer of $J_{\lambda}$ and a critical point of $J_{\lambda}$, thus by Proposition 3.1-(3) $u$ is a critical point of $I_{\lambda}$, as required.

Let $\lambda_{i}\in(0, 1]$ with $\lambda_{i}\to0$ as $i\rightarrow \infty$. Let $\{u_{i}\}\subset E$ be a sequence of critical points of $I_{\lambda_{i}}$ with $I_{\lambda_{i}}(u_i) = c_{\lambda_{i}}\leq C$. According to Lemma 3.4, there exists a critical point $\widetilde{u}$ of $I$ such that $\widetilde{u}\in H^{1}_{V}(\mathbb{R}^3)\cap L^{\infty}(\mathbb{R}^3)$. In the following, we will show that $\widetilde{u}$ is a non-trivial critical point of $I$. Considering $\langle I_{\lambda_i}'(u_i), u_i\rangle = 0$, it follows from Sobolev inequality, interpolation inequality, and Young's inequality that

which implies $\|u_{i}\|_{p}\geq(\frac{C_{11}}{C_{12}})^{1/(p-4)}$. Recall that $u_{i}\to \widetilde{u}$ strongly in $L^{p}(\mathbb{R}^{3})$ for $4\leq p < 12$. Therefore, we see that $\widetilde{u}\neq0$.

4.   Proof of Theorem 1.2

The proof of Theorem 1.2 is similar to that made in Section 3. From Lemmas 3.1 and 3.2, it is clear that the functional $I_{\lambda}$ on $\mathcal{N}_{\lambda}$ has a bounded minimizing sequence $\{u_n\}$. But we cannot ensure this sequence to be convergent in $E^{\ast}: = W^{1, 4}(\mathbb{R}^{3})\cap H^{1}(\mathbb{R}^{3})$, which endowed with the norm

Thus, we need to study some compact properties of the minimizing sequence for $I_{\lambda}$ on the Nehari manifold $\mathcal{N}^*_{\lambda}$, where

Firstly, we have the following result due to P.L. Lions (^[32]):

Lemma 4.1. Let $r > 0$. If $\{u_n\}$ is bounded in $E^{\ast}$ and

we have $u_n\to0$ strongly in $L^{s}(\mathbb{R}^{3})$ for any $s\in(2, 12)$.

Next, we are going to discuss the minimizing sequence for $I_{\lambda}$ on $\mathcal{N}^*_{\lambda}$.

Lemma 4.2. Let $\{u_n\}\subset\mathcal{N}^*_{\lambda}$ be a minimizing sequence for $I_{\lambda}$. Then $\{u_n\}$ is bounded in $E^{\ast}$. Moreover, after a suitable $\mathbb{Z}^{3}$-translation, passing to a subsequence there exists $u\in\mathcal{N}^*_{\lambda}$ such that $u_n\rightharpoonup u$ and $I_{\lambda}(u) = \inf\limits_{\mathcal{N}^*_{\lambda}}I_{\lambda}$.

Proof. Set $c = \inf\limits_{\mathcal{N}^*_{\lambda}}I_{\lambda}$. Remind that $\{u_n\}$ is bounded by Lemma 3.2, $u_n\rightharpoonup u$ weakly in $E^{\ast}$ after passing to a subsequence. If

then $u_n\to0$ strongly in $L^{s}(\mathbb{R}^3)$ for any $s\in(2, 12)$ due to Lemma 4.1. Then, by (1.3) it is easy to see that

Therefore,

which implies $\|u_n\|_{W}\to0$. This contradicts with Lemma 3.1-(2). Hence, there exist $r, \delta > 0$ and a sequence $\{y_n\}\subset\mathbb{R}^3$ such that

where we may assume $y_n\in\mathbb{Z}^3$. Due to the invariance of $I_{\lambda}$ on $\mathcal{N}^*_{\lambda}$ under translations, $\{y_n\}$ is bounded in $\mathbb{Z}^{3}$. Hence, passing to a subsequence we imply $u_n\rightharpoonup u\neq0$ weakly in $E^*$ and $I'_{\lambda}(u) = 0$. It follows that $u\in\mathcal{N}^*_{\lambda}$, and then $I_{\lambda}(u)\geq c > 0$.

From (1.4), Fatou's lemma and the weakly lower semi-continuity of norm, we have

which implies $I_{\lambda}(u)\leq c$. This completes the proof.

Proof of Theorem 1.2 Combining Lemma 4.2 and the methods in proving Theorem 1.1, we can prove that the conclusion of Theorem 1.2 is true.

5.   Proof of Theorem 1.3

In this section, we firstly need to consider the associated "limit problem" of (1.8):

where $a > 0, b\geq0$, $V_{\infty}$ is defined as shown in $(V^*_2)$.

Since problem (5.1) involves the quasilinear term $u\Delta (u^{2})$ and the nonlocal term, its natural energy functional is not well defined in $H^{1}_V({\mathbb{R}}^{3})$. To solve this difficulty, we set

In addition, for convenience, we make use of the following notations:

$\bullet \; H_{r}^{1}(\mathbb{R}^{3}): = \left\{u: u \in \widetilde E, u(x) = u(|x|)\right\}$;

$\bullet \; P: = \left\{u \in \widetilde E | u \geq 0\right\}$ denotes the positive cone of $\widetilde E$ and $P_{+} = P \backslash\{0\}$;

$\bullet \; u^{+}: = \max \{u, 0\}$ and $u^{-} = \min \{u, 0\}$;

$\bullet$ For any $u \in\widetilde E \backslash\{0\}, u_{t}$ is defined as

Now we give some preliminary results as follows.

Lemma 5.1. Assume $f\in C^1({\mathbb{R}}^+, {\mathbb{R}})$ satisfies $(f^*_1)-(f^*_4)$, then

(i) For every $\varepsilon > 0$ and $p\in (2, 12)$, there is $C_{\varepsilon} > 0$ such that

(ii) $F(s) > 0, sf(s) > 2F(s)$ and $sf^{\prime}(s) > f(s)$ if $s > 0$.

Proof. It is easy to get the results by direct calculation, so we omit the proof.

Lemma 5.2. (Pohozaev identity, ^[33]) Assume that $(f^*_1)-(f^*_4)$ hold. If $u\in \widetilde E$ is a weak solution to equation (5.1), then the following Pohozaev identity holds:

Proof. The proof is standard, so we omit it.

Lemma 5.3. Assume that $(f^*_3)$ holds. Then the functional

is not bounded from below.

Proof. For any $u\in P_{+}$, we obtain

By $(f_{3}^{*})$, it is clear that $I_{V_\infty}(u_t)\rightarrow -\infty$ as $t\rightarrow +\infty$.

Lemma 5.4. Let $C_{13}, C_{14}, C_{15}$ be positive constants and $u\in P_{+}$. If $f\in C^1$ satisfies $(f^*_1)-(f^*_4)$, then the function

has a unique positive critical point which corresponds to its maximum.

Proof. The conclusion is easily obtained by elementary calculation.

Now set

where $P(u)$ is given by (5.3). Then, by direct calculation we have

Lemma 5.5. For any $u\in P_{+}$, there exists a unique $\widetilde{t} > 0$ such that $u_{\widetilde{t}}\in \mathcal{M}$. Moreover, $I_{V_{\infty}}(u_{\widetilde{t}}) = \max\limits_{t > 0}I_{V_{\infty}}(u_{t})$.

Proof. For any $u\in P_{+}$ and $t > 0$, let $\gamma(t): = I_{V_{\infty}}(u_{t})$. By Lemma 5.4, $\gamma(t)$ has a unique critical point $\widetilde{t} > 0$ corresponding to its maximum, i.e., $\gamma({\widetilde{t}}) = \max\limits_{t > 0}\gamma(t)$ and $\gamma^{\prime}({\widetilde{t}}) = 0$. It follows that $G(u_{\widetilde{t}}) = \widetilde{t}\gamma^{\prime}({\widetilde{t}}) = 0$. Thus, $u_{\widetilde t}\in \mathcal{M}$.

We define

and

where $u_{t}(x)$ is given by (5.2) and

Lemma 5.6. $z_1 = z_{2} = z_{3} = z_{4} > 0$.

Proof. We divide the proof into the following three steps:

Step 1. $z_3 > 0$. For any $u\in \mathcal{M}$, by Lemma 5.1-(i), the continuous embedding $\widetilde E\hookrightarrow L^{s}({\mathbb{R}}^{3})$ for $s\in[2, 12)$ and Sobolev inequality, we get

where $C_{\varepsilon} > 0$ is a constant depending on $\varepsilon$. Since $u\neq 0$ and $p > 2$, then for $\varepsilon, t > 0$ small enough, we deduce $I_{V_{\infty}}(u) > 0$. Furthermore, we get $z_3 > 0$.

Step 2. $z_1 = z_2 = z_3$. The proof is similar to the argument of Nehari manifold method in ^[34]. One can make obvious modification by Lemma 5.4 and 5.5.

Step 3. $z_3 = z_4$. Since equation (5.1) is autonomous, the proof is standard by Schwartz symmetric arrangement.

In the following discussion, for convenience, we set $z = z_1( = z_2 = z_3 = z_4)$.

Lemma 5.7. If $z$ is attained at some $u\in \mathcal{M}$, then $u$ is a critical point of $I_{V_{\infty}}$ in $\widetilde E$.

Proof. Since this proof is analogous to the proof of Lemma 2.7 in ^[11], we omit it.

Lemma 5.8. Assume $(f^{*}_1)-(f^{*}_4)$ hold. Then problem (5.1) has a positive ground state solution.

Proof. From Lemma 5.6 and Lemma 5.7, we only need to prove that $z$ is achieved for some $u \in H_{r}^{1}(\mathbb{R}^{3}) \cap \mathcal{M}$.

Letting $\left\{u_{n}\right\} \subset H_{r}^{1}(\mathbb{R}^{3}) \cap \mathcal{M}$ be a minimizing sequence of $I_{V_{\infty}},$ then we have

for $n$ large enough. Therefore, $\left\{\|\nabla u_{n}\|_{2}^{2}\right\}$ and $\left\{\|\nabla (u_{n}^{2})\|_{2}^{2}\right\}$ are bounded. In the following we prove $\left\{\|u_{n}\|^{2}_2\right\}$ is also bounded. By $u_{n} \in \mathcal{M}$ and Lemma 5.1-(ii) we obtain

where $q\in(2, 12).$ According to the interpolation and Sobolev inequalities, we have

where $\frac{1}{q} = \frac{\theta}{2}+\frac{1-\theta}{12}$. Noting $q \theta < 2$, by Young's inequality, we derive for some $C_{\varepsilon}^{\prime} > 0$

Hence we obtain $\left\{\|u_{n}\|_{2}^{2}\right\}$ is also bounded if we pick $\varepsilon = \frac{1}{2} V_{\infty}.$ Therefore, $\left\{u_{n}\right\}$ is bounded in $\widetilde E$.

Recall the compact embedding $H_{r}^{1}(\mathbb{R}^{3}) \hookrightarrow L^{p}(\mathbb{R}^{3})$ for $p \in(2, 12)$. Thus, going if necessary to a subsequence, we may assume that there exists a function $u \in \widetilde E$ such that

It is easy to check $u^{+} \neq 0$ and $G(u) \leq 0.$ By Lemma 5.5, $u_{t_{0}} \in \mathcal{M}$ for some $0 < t_{0} \leqslant 1.$ If $t_{0} \in(0, 1)$, one can easily verify $I_{V_{\infty}}\left(u_{t_{0}}\right) < z.$ Hence $t_{0} = 1$ and $z$ is attained at some $u \in \mathcal{M}$.

The strong maximum principle and standard argument ^[35] imply that $u(x)$ is positive for all $x \in \mathbb{R}^{3}$. Therefore, $u$ is a positive ground state solution of problem (5.1).

So far, we have proved that the associated "limit problem" of (1.8) has a ground state solution. Next, on this basis, we are going to prove Theorem 1.3.

Since $V$ is not a constant, that is to say, problem (1.8) is no longer autonomous, the method to prove Lemma 5.8 cannot be applied. Moreover, due to the lack of the variant Ambrosetti-Rabinowitz condition, we could not obtain the boundedness of any $(P S)_{c}$ sequence. In order to overcome this difficulty, we make use of the monotone method due to L. Jeanjean.

Proposition 5.1. (^[36], Theorem 1.1) Let $\left(\widetilde E, \|\cdot\|_{H_V}\right)$ be a Banach space and $T \subset \mathbb{R}^{+}$ be an interval. Consider a family of $C^{1}$ functionals on $\widetilde E$ of the form

with $B(u) \geqslant 0$ and either $A(u) \rightarrow+\infty$ or $B(u) \rightarrow+\infty$ as $\|u\|_{H_V} \rightarrow+\infty.$ Assume that there are two points $v_{1}, v_{2} \in \widetilde E$ such that

where

Then, for almost every $\lambda \in T,$ there is a bounded $(PS)_{c_{\lambda}}$ sequence in $\widetilde E$.

Letting $T = [\delta, 1],$ where $\delta \in(0, 1)$ is a positive constant, we investigate a family of functionals on $\widetilde E$ with the following form

Then let $I_{V, \lambda}(u) = A(u)-\lambda B(u),$ where

and

It is easy to see that $A(u)\rightarrow \infty$ as $\|u\|_{H_V}\rightarrow \infty$ and $B(u)\geq 0$.

Lemma 5.9. Under the assumptions of Theorem 1.3 we have

(i) there exists $v \in \widetilde E \backslash\{0\}$ such that $I_{V, \lambda}(v) \leqslant 0$ for all $\lambda \in[\delta, 1]$;

(ii) $c_{\lambda} = \inf\limits _{\gamma \in \Gamma} \max\limits _{t \in[0, 1]} I_{V, \lambda}(\gamma(t)) > \max \left\{I_{V, \lambda}(0), I_{V, \lambda}(v)\right\}$ for all $\lambda \in[\delta, 1],$ where

Proof. $(i)$ For any $\lambda \in[\delta, 1]$, $t > 0$ and $u\in P_+$, we get

Then by $(f^*_3)$, we infer that there exists $t > 0$ such that $I_{V, \lambda}\left(u_{t}\right) \leq I_{V_{\infty}, \delta}\left(u_{t}\right) < 0$.

$(ii)$ Depending on Lemma 5.1-$(i)$, for $\varepsilon > 0$ small enough and $p \in(2, 12),$ there exists $C_{\varepsilon} > 0$ such that

Then by standard argument there exists $r > 0$ such that

and hence $c_{\lambda} > \max \left\{I_{V, \lambda}(0), I_{V, \lambda}\left(u_{t}\right)\right\}.$ Then the conclusion follows with $v = u_{t}$.

Lemma 5.10. (^[36], Lemma 2.3) Under the assumptions of Proposition 5.1, the map $\lambda \rightarrow c_{\lambda}$ is non-increasing and left continuous.

By Lemma 5.8, we infer that for any $\lambda \in[\delta, 1],$ the "limit problem'' of the following type:

has a positive ground state solution in $\widetilde E.$ Thus we further derive that for any $\lambda \in[\delta, 1],$ there exists

such that $u_{\lambda}(x) > 0$ for all $x \in \mathbb{R}^{3}, \; I_{V_{\infty}, \lambda}^{\prime}\left(u_{\lambda}\right) = 0$ and

where

Lemma 5.11. Suppose that $(V^{*}_{1})-(V^{*}_{2})$, $(f^{*}_{1})-(f^{*}_{4})$ hold and $V(x) \not\equiv V_{\infty}.$ Then there exists $\bar{\lambda}\in[\delta, 1)$ such that $c_{\lambda} < m_{\lambda}$ for any $\lambda \in[\bar{\lambda}, 1]$.

Proof. First of all, for convenience, we set $I_{V, \lambda}(u) = I_{V, 1}(u)$, $m_\lambda = m_1$ and $c_{\lambda} = c_1$ when $\lambda = 1$. And let $u_\lambda, u_1$ be the minimizer of $I_{V, \lambda}, I_{V, 1}$, respectively. By Lemma 5.3, we see that there exists $K > 0$ independent of $\lambda$ such that $I_{V, \lambda}((u_1)_{K}) < 0$ for all $\lambda\in[\delta, 1]$. Moreover, It is easy to see that $I_{V, \lambda}\left(\left(u_{1}\right)_{t}\right)$ is continuous on $t \in[0, \infty).$ Hence for any $\lambda \in \; [\delta, 1),$ we can choose $t_{\lambda} \in(0, K)$ such that $I_{V, \lambda}\left(\left(u_{1}\right)_{t_{\lambda}}\right) = \max\limits _{t \in[0, K]} I_{V, \lambda}\left(\left(u_{1}\right)_{t}\right)$. Note that $I_{V, \delta}\left((u_{1})_{t}\right) \rightarrow-\infty$ as $t \rightarrow \infty,$ thus there exists $K_{0} > 0$ such that

By the definition of $t_{\lambda},$ one has

Then the above two inequalities implies $t_{\lambda} < K_{0}$ for $\lambda \in[\delta, 1].$ Let $\beta_{0} = \inf\limits _{\lambda \in[\delta, 1]} t_{\lambda}.$ If $\beta_{0} = 0,$ then by contradiction, there exists a sequence $\left\{\lambda_{n}\right\} \subset[\delta, 1]$ such that $\lambda_{n} \rightarrow \lambda_{0} \in[\delta, 1] \; \text { and }\; t_{\lambda_{n}} \rightarrow 0.$ It follows that

which implies $\beta_{0} > 0.$ Thus

Let

Then $\delta\leq \bar{\lambda} < 1.$ From the definition of $\bar{\lambda}$ and $0 < \beta_{0} \leq t_{\lambda} < K_{0}$ for $\forall \lambda \in[\delta, 1]$, we have

Next, we will introduce the following global compactness lemma, which is used for proving that the functional $I_{V, \lambda}$ satisfies $(PS)_{c_{\lambda}}$ condition for all $\lambda \in[\bar{\lambda}, 1]$.

Lemma 5.12. Suppose that $(V^{*}_{1})-(V^{*}_{2})$ and $(f^{*}_{1})-(f^{*}_{4})$ hold. For $c > 0$ and $\lambda \in[\delta, 1]$, let $\left\{u_{n}\right\} \subset \widetilde E$ be a bounded $(PS)_{c}$ sequence for $I_{V, \lambda}$. Then there exists $v_{0} \in \widetilde E$ and $A \in \mathbb{R}$ such that $J_{V, \lambda}^{\prime}\left(v_{0}\right) = 0,$ where

Moreover, there exists a finite (possibly empty) set $\left\{v_{1}, \ldots, v_{l}\right\} \subset \widetilde E$ of nontrivial solutions for

and $\left\{y_{n}^{k}\right\} \subset \mathbb{R}^{3}$ for $k = 1, \ldots, l$ such that

Proof. The proof is analogous to Lemma 5.3 in ^[10]. Here we only point out the difference. Since $f$ satisfies $(f^{*}_{1})-(f^{*}_{4}),$ for $u_{n} \rightharpoonup u$ in $\widetilde E$, we have

where ${\widetilde E}{'}$ is the conjugate space of ${\widetilde E}$. Moreover, by referring to Lemma 3.4-(12) in ^[23], we can get

Then the rest proof can be derived by obvious modification from line to line.

Lemma 5.13. Suppose that $(V^{*}_{1})-(V^{*}_{2})$ and $(f^{*}_{1})-(f^{*}_{4})$ hold. For $\lambda \in[\bar{\lambda}, 1],$ let $\left\{u_{n}\right\} \subset \widetilde E$ be a bounded $(P S)_{c_ \lambda}$ sequence of $I_{V, \lambda}.$ Then there exists a nontrivial $u_{\lambda} \in \widetilde E$ such that

Proof. According to Lemma 5.12 and referring to the proof of Lemma 3.5 in ^[10], we can easily complete this proof. So we omit the detailed proof.

In order to prove that the problem (1.8) has a positive ground state solution, we define

where $\mathcal{X} : = \left\{u \in \widetilde E \backslash\{0\}: I_{V}^{\prime}(u) = 0\right\}$.

Lemma 5.14. $\mathcal{X}\neq \emptyset$.

Proof. Depending on Lemma 5.9 and Proposition $5.1,$ we see for almost everywhere $\lambda \in[\bar{\lambda}, 1],$ there exists a bounded sequence $\left\{u_{n}\right\} \subset \widetilde E$ such that

It follows from Lemma 5.13 that $I_{V, \lambda}$ has a nontrivial critical point $u_{\lambda} \in \widetilde E$ and $I_{V, \lambda}\left(u_{\lambda}\right) = c_{\lambda}$.

Based on the above discussion, there exists a sequence $\left\{\lambda_{n}\right\} \subset[\bar{\lambda}, 1]$ with $\lambda_{n} \rightarrow 1^{-}$ and an associated sequence $\left\{u_{\lambda_{n}}\right\} \subset \widetilde E$ such that $I_{V, \lambda_{n}}\left(u_{\lambda_{n}}\right) = c_{\lambda_{n}}, I_{V, \lambda_{n}}^{\prime}\left(u_{\lambda_{n}}\right) = 0$.

Next, we prove that $\left\{u_{\lambda_{n}}\right\}$ is bounded in $\widetilde E.$ By $(V^{*}_{1})$ and Hardy inequality, using the proof of Lemma 5.8, we can refer that $\left\{\|\nabla u_{\lambda_{n}}\|_{2}\right\}$ and $\left\{\|u_{\lambda_{n}}\|_{2}\right\}$ are bounded. Thus, $\left\{u_{\lambda_{n}}\right\}$ is bounded in $\widetilde E$.

Since $\lambda_{n} \rightarrow 1^{-},$ we claim that $\left\{u_{\lambda_{n}}\right\}$ is a $(P S)_{c_{1}}$ sequence of $I_{V} = I_{V, 1}.$ Indeed, by Lemma 5.10 we obtain that

and for all $\varphi \in H^{1}(\mathbb{R}^{3}) \backslash\{0\}$,

Hence $\left\{u_{\lambda_{n}}\right\}$ is a bounded $(P S)_{c_{1}}$ sequence of $I_{V}$. Then by Lemma 5.13, $I_{V}$ has a nontrivial critical point $u_{0} \in \widetilde E$ and $I_{V}\left(u_{0}\right) = c_{1}$. Thus, $\mathcal{X}\neq \emptyset$.

Proof of Theorem 1.3 Firstly, in order to get a nontrivial $(P S)_{m}$ sequence, we need to prove $m > 0$.

For all $u \in \mathcal{X},$ we have $\langle I_{V}^{\prime}(u), u\rangle = 0$. Thus by standard argument we see $\|u\|_{H_V} \geq \xi$ for some positive constant $\xi.$ On the other hand, the Pohozaev identity (5.3) holds, i.e., $P_{V}(u) = 0$. Now by Lemma 5.1-(ⅱ) we can get

Then from $(V^{*}_{1})$ and Hardy inequality, we infer

Therefore, we obtain $m \geq 0$.

In the following let us rule out $m = 0$. By contradiction, let $\left\{u_{n}\right\}$ be a $(P S)_{0}$ sequence of $I_{V}.$ Then it is easy to show that $\lim \limits_{n \rightarrow \infty}\left\|u_{n}\right\|_{H_V} = 0,$ which contradicts with $\left\|u_{n}\right\|_{H_V} \geq \xi > 0$ for all $n \in \mathbb{N}$.

Next, we may assume that there exists a sequence $\left\{u_{n}\right\} \subset P_{+}$ satisfying $I_{V}^{\prime}\left(u_{n}\right) = 0$ and $I_{V}\left(u_{n}\right) \rightarrow m.$ Similar to the argument in the proofs of Lemma 5.14, we can conclude that $\left\{u_{n}\right\}$ is a bounded $(P S)_{m}$ sequence of $I_{V}$. Then by Lemma 5.13 and strong maximal principle, there exists a function $u\in\widetilde E$ such that

So $u$ is a positive ground state solution for problem (1.8). The proof is completed.

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.