# Ground state solutions of Nehari–Pohožaev type for a kind of nonlinear problem with general nonlinearity and nonlocal convolution term

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## Abstract

In this paper, we consider the following nonlinear problem with general nonlinearity and nonlocal convolution term:

$$\textstyle\begin{cases} -\Delta u+V(x)u+(I_{\alpha }\ast \vert u \vert ^{q}) \vert u \vert ^{q-2}u=f(u), \quad x\in {\mathbb{R}}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}), \quad \end{cases}$$

where $$a\in (0,3)$$, $$q\in [1+\frac{\alpha }{3},3+\alpha )$$, $$I_{\alpha }:\mathbb{R}^{3}\rightarrow \mathbb{R}$$ is the Riesz potential, $$V\in \mathcal{C}(\mathbb{R}^{3},[0,\infty ))$$, $$f\in \mathcal{C}(\mathbb{R},\mathbb{R})$$ and $$F(t)=\int _{0}^{t}f(s)\,ds$$ satisfies $$\lim_{|t|\to \infty }F(t)/|t|^{\sigma }=\infty$$ with $$\sigma =\min \{2,\frac{2\beta +2}{\beta }\}$$ where $$\beta =\frac{ \alpha +2}{2(q-1)}$$. By using new analytic techniques and new inequalities, we prove the above system admits a ground state solution under mild assumptions on V and f.

## Introduction

In this paper we are concerned with the existence of ground state solutions for the nonlinear problem:

$$\textstyle\begin{cases} -\Delta u+V(x)u+(I_{\alpha }\ast \vert u \vert ^{q}) \vert u \vert ^{q-2}u=f(u), & x\in {\mathbb{R} }^{3}, \\ u\in H^{1}(\mathbb{R}^{3}), \end{cases}$$
(1.1)

where $$\alpha \in (0,3)$$, $$q\in [1+\frac{\alpha }{3},3+\alpha )$$, $$V\in \mathcal{C}(\mathbb{R}^{3},[0,\infty ))$$, $$f\in \mathcal{C}( \mathbb{R},\mathbb{R})$$ and $$I_{\alpha }:\mathbb{R}^{3}\rightarrow \mathbb{R}$$ is the Riesz potential of order $$\alpha \in (0,3)$$, given by

$$I_{\alpha }(x)=\frac{A_{\alpha }}{ \vert x \vert ^{N-\alpha }} \quad \mbox{with } A_{\alpha }= \frac{\varGamma (\frac{N-\alpha }{2})}{\varGamma (\frac{\alpha }{2})\pi ^{N/2}2^{\alpha }}, x\in \mathbb{R}^{3}\setminus \{0\}.$$

When $$q=2$$ and $$\alpha =2$$, the convolution term of (1.1) can be considered as analogous to the internal potential ϕ in Schrödinger–Poisson system like the following equation:

$$\textstyle\begin{cases} -\Delta u +V(x)u +\phi u=f(u), & x\in \mathbb{R}^{3}, \\ -\Delta \phi =u^{2}, & x\in \mathbb{R}^{3}. \end{cases}$$
(1.2)

System (1.2) was first introduced in  as a model describing solitary waves for the nonlinear stationary Schrödinger equation interacting with the electrostatic field. We note that system (1.2) is also called a Schrödinger–Maxwell equation system, for more details on the physical aspects of this problem, we refer to  and the references therein.

In recent years, the existence, multiplicity and concentration of nontrivial solutions of (1.2) have been the subject of extensive mathematical studies, for example, [2,3,4,5,6,7,8]. Problem (1.2) with $$V(x)\equiv 1$$ or being radially symmetric, has been widely studied under various conditions on f; see for example [9,10,11,12,13,14,15,16,17]. When $$V=1$$ and $$f(u)=|u|^{p-2}u$$, by introducing a new manifold that is defined by a condition which is a combination of the Nehari equation and the Pohožaev equality, Ruiz  showed that (1.2) admits a positive radial solution if $$3< p<6$$, but does not have a nontrivial solution for $$2< p\leq 3$$. Under the same assumptions, based on Ruiz’s approach in , Azzollini and Pomponio  obtained the existence of ground state solutions for (1.2) by using a concentration-compactness argument. When $$f(u)=|u|^{p-2}u$$ and V satisfies the following assumptions:

(V1):

$$V\in \mathcal{C}(\mathbb{R}^{3},[0,\infty ))$$ and $$V_{\infty }:=\lim_{|y|\rightarrow \infty }V(y)\geq V(x)$$ for $$x\in \mathbb{R}^{3}$$;

(V2′):

$$V(x)$$ is weakly differentiable, and satisfies $$\nabla V(x) \cdot x \in L^{\infty }(\mathbb{R}^{3})\cup L^{3/2}( \mathbb{R}^{3})$$, and

$$2V(x)+\nabla V(x) \cdot x \geq 0 \quad \mbox{a.e. } x\in \mathbb{R}^{3},$$
Zhao and Zhao  established the existence of ground state solutions for (1.2) by using the Jeanjean’s monotonicity trick . In a recent paper , Tang and Chen introduced some new tricks to generalize and improve the results in [3, 8, 16] to the more general case where V satisfies (V1) and (V2′) and f satisfies the following assumptions:

1. (F1)

$$f\in \mathcal{C}(\mathbb{R},\mathbb{R})$$, and there exist constants $$C>0$$ and $$q\in (2,6)$$ such that

$$\bigl\vert f(t) \bigr\vert \leq C\bigl(1+ \vert t \vert ^{q-1} \bigr), \quad \forall t\in \mathbb{R};$$
2. (F2)

$$f(t)=o(t)$$ as $$t\rightarrow 0$$;

3. (F3)

$$\lim_{|t|\rightarrow \infty }\frac{F(t)}{|t|^{3}}=\infty$$, where $$F(t)=\int _{0}^{t}f(s)\,ds$$;

4. (F4)

$$[2f(t)t-3F(t)]/t^{3}$$ is nondecreasing on $$(-\infty ,0) \cup (0,+\infty )$$.

There is also other work about ground state solutions for (1.2); we refer to [20, 21]. Motivated by the above work and [22,23,24,25,26,27,28,29,30], in the present paper, we shall extend the results concerning the existence of ground state solutions for (1.2) in  to (1.1). Compared with (1.2), it is more difficult to deal with (1.1) for the reason that $$q \in [1+\frac{ \alpha }{3},3+\alpha )$$. Because of the changing of q, the competing effect of $$(I_{\alpha }\ast |u|^{q})|u|^{q-2}u$$ and $$f(u)$$ is also changing.

For any $$\varepsilon >0$$, it follows from (F1) and (F2) that there exists $$C_{\varepsilon } >0$$ such that

$$\bigl\vert f(t) \bigr\vert \leq \varepsilon \vert t \vert +C_{\varepsilon } \vert t \vert ^{p-1}, \quad \forall t \in \mathbb{R}.$$
(1.3)

Under assumptions (V1), (F1), (F2) and (1.3), the functional

$$\varPhi (u)=\frac{1}{2} \int _{\mathbb{R}^{3}}\bigl[ \vert \nabla u \vert ^{2}+V(x)u^{2} \bigr]\,dx+ \frac{1}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}\,dx- \int _{\mathbb{R}^{3}}F(u)\,dx$$
(1.4)

is well defined in $$H^{1}(\mathbb{R}^{3})$$ and $$\varPhi \in \mathcal{C} ^{1}(H^{1}(\mathbb{R}^{3}),\mathbb{R})$$, where $$F(t)=\int _{0}^{t}f(s) \,ds$$. Moreover, for any $$u,\upsilon \in H^{1}(\mathbb{R}^{3})$$,

\begin{aligned} \bigl\langle \varPhi ^{\prime }(u),v\bigr\rangle =& \int _{\mathbb{R}^{3}}\bigl[\nabla u\cdot \nabla v+V(x)uv\bigr]\,dx \\ &{}+ \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q-2}uv\,dx- \int _{\mathbb{R}^{3}}f(u)v\,dx. \end{aligned}
(1.5)

Hence, the solution of (1.1) are critical points of $$\varPhi (u)$$. A solution is called a ground state solution if its energy is minimal among all nontrivial solutions.

In this paper, let $$\beta =\frac{\alpha +2}{2(q-1)}$$, and in addition to (V1), (F1) and (F2), we also need to introduce the following assumptions:

(V2):

$$V\in \mathcal{C}^{1}(\mathbb{R}^{3})$$, the set $$\{x\in \mathbb{R}^{3}:|\nabla V(x)\cdot x|\geq \epsilon \}$$ has finite Lebesgue measure for every $$\epsilon >0$$, and the function $$t\mapsto t^{2}[(2\beta -3)V(tx)-\nabla V(tx)\cdot (tx)]$$ is increasing on $$(0,+\infty )$$ for every $$x\in \mathbb{R}^{3}$$;

(V3):

$$V\in \mathcal{C}^{1}(\mathbb{R}^{3})$$, $$\nabla V(x) \cdot x \in L^{\infty }(\mathbb{R}^{3})$$ and there exists $$\varrho >0$$ such that

$$2V(x)+\nabla V(x)\cdot x\geq \varrho , \quad \forall x\in \mathbb{R}^{3};$$
(V3′):

$$V\in \mathcal{C}^{1}(\mathbb{R}^{3})$$, $$\nabla V(x) \cdot x \in L^{\infty }(\mathbb{R}^{3})$$, $$2V(x)+\nabla V(x)\cdot x \geq \varrho$$, $$\forall \ x\in \mathbb{R}^{3}$$ and there exists $$\mu > \frac{2\beta +2}{\beta }$$ such that

$$f(t)t-\mu F(t)\geq 0,\quad \forall t\in \mathbb{R} ;$$
(F5):

$$\lim_{|t|\rightarrow \infty } \frac{F(t)}{|t|^{\frac{2 \beta +2}{\beta }}}=\infty$$;

(F6):

$$[\beta f(t)t-3F(t)]/t|t|^{\frac{\beta +2}{\beta }}$$ is nondecreasing on both $$(-\infty ,0)$$ and $$(0,+\infty )$$.

To state our results, we define the Nehari–Pohožaev manifold as follows:

$$\mathcal{M}:=\bigl\{ {u\in H^{1}\bigl( \mathbb{R}^{3}\bigr)\setminus \{0\}}:J(u):= \beta \bigl\langle \varPhi '(u),u\bigr\rangle -\mathcal{P}(u)=0\bigr\} ,$$
(1.6)

where

\begin{aligned} \mathcal{P}(u) = &\frac{1}{2} \Vert \nabla u \Vert _{2}^{2}+\frac{1}{2} \int _{\mathbb{R}^{3}}\bigl[3V(x)+\nabla V(x)\cdot x\bigr]u^{2} \,dx +\frac{3+ \alpha }{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}\,dx \\ &{}-3 \int _{\mathbb{R}^{3}}F(u)\,dx \end{aligned}
(1.7)

which is associated with the Pohožaev identity $$\mathcal{P}(u)=0$$ of (1.1) that can be obtained by the same argument as in [13, 31]. And

\begin{aligned} J(u) = &\frac{2\beta -1}{2} \Vert \nabla u \Vert _{2}^{2}+\frac{1}{2} \int _{\mathbb{R}^{3}}\bigl[(2\beta -3)V(x)-\nabla V(x)\cdot x \bigr]u^{2}\,dx \\ &{}+\frac{2\beta -1}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}\,dx - \int _{\mathbb{R}^{3}}\bigl[\beta f(u)u-3F(u)\bigr]\,dx. \end{aligned}
(1.8)

Throughout the paper we use the following notations:

• $$H^{1}(\mathbb{R}^{3})$$ denotes the usual Sobolev space equipped with the inner product and norm

$$(u,v)= \int _{\mathbb{R}^{3}}(\nabla u\cdot \nabla v+uv)\,dx, \quad \Vert u \Vert =(u,u)^{\frac{1}{2}},\quad \forall u,v\in H^{1}\bigl( \mathbb{R}^{3}\bigr).$$
• $$L^{s}(\mathbb{R}^{3})(1\leq s <\infty )$$ denotes the Lebesgue space with the norm $$\|u\|_{s}=(\int _{\mathbb{R}^{3}}|u|^{s}\,dx)^{1/s}$$;

• for any $$u\in H^{1}(\mathbb{R}^{3}) \setminus \{0\}$$, $$u_{t}(x):=u(tx)$$ for $$t > 0$$;

• for any $$x\in \mathbb{R}^{3}$$ and $$r > 0$$, $$B_{r}(x):=\{y\in \mathbb{R}^{3}:|y-x| < r\}$$;

• $$C_{1},C_{2},\ldots$$ denotes positive constants possibly different in different places.

Next,we state the main results of this paper.

### Theorem 1.1

Assume that V and f satisfy (V1), (V2), (F1), (F2), (F5) and (F6) hold. Then problem (1.1) has a ground state solution $$u_{0}\in H^{1}(\mathbb{R}^{3})$$ such that $$\varPhi (u_{0})= \inf_{\mathcal{M}}\varPhi =\inf_{u\in H^{1}(\mathbb{R}^{3})\setminus \{0\}}\max_{t > 0} \varPhi (t^{\beta }u_{t}) > 0$$.

### Theorem 1.2

Assume that V and f satisfy (V1), (V3), (F1), (F2), (F5) and (F6) hold. Then problem (1.1) has a positive ground state solution.

### Theorem 1.3

Assume that V and f satisfy (V1), (V3′), (F1), (F2), (F5) and (F6) hold. Then problem (1.1) has a positive ground state solution.

Inspired by [19, 32], we shall prove Theorem 1.1 following this scheme:

1. Step (i)

we verify $$\mathcal{M}\neq \emptyset$$ and establish the minimax characterization of $$m=\inf_{\mathcal{M}} \varPhi >0$$;

2. Step (ii)

we prove that m can be obtained;

3. Step (iii)

we show that the minimizer of Φ on $$\mathcal{M}$$ is a critical point.

Although we mainly follow the procedure of [19, 32], we have to face many new difficulties due to the mutual competing effect between $$f(u)$$ and $$(I_{\alpha }\ast |u|^{q})|u|^{q-2}u$$. More precisely, in Step (i), we first establish a key inequality in Lemma 2.2 by using some properties about nonlinearity term, and also we use Nehari–Pohožaev manifold to remove the influence of the term $$(I_{\alpha }\ast |u|^{q})|u|^{q-2}u$$ which needs a kind of computational technique; it is worth mentioning that [23, Proposition 2.7] gives us an excellent numerical property as regards $$V(x)$$ which is convenient for us to eliminate the potential term $$V(x)$$ in some inequalities. Then we construct a saddle point structure with respect to the fibre $$\{t^{\beta }u_{t}:t>0\} \subset H^{1}(\mathbb{R}^{3})$$ for $$u\in H^{1}(\mathbb{R}^{3}) \setminus \{0\}$$; see Lemma 2.6, finally based on these constructions we obtain the minimax characterization of m; see Lemma 2.8. In Step (ii), we first choose a minimizing sequence $$\{u_{n}\}$$ of Φ on $$\mathcal{M}$$, and show that $$\{u_{n}\}$$ is bounded in $$H^{1}(\mathbb{R}^{3})$$, then with the help of the key inequality established in Lemma 2.2 and a concentration-compactness argument, we prove that there exist $$\hat{u} \in H^{1}(\mathbb{R}^{3})$$ and $$\hat{t} >0$$ such that $$u_{n}\rightharpoonup \hat{u}$$ in $$H^{1}(\mathbb{R}^{3})$$ up to translations and extraction of a subsequence, and $$\hat{t}^{\beta }\hat{u}_{\hat{t}} \in \mathcal{M}$$ is a minimizer of $$\inf_{\mathcal{M}} \varPhi$$; see Lemma 2.13 and Lemma 2.14. Step (iii) is similar to [19, Lemma 2.10].

Motivated by [8, 19], we use the Jeanjean’s monotonicity trick  to prove Theorems 1.2 and 1.3, which can helps us to construct a bounded (PS) sequence. The difficulty in the proof is to overcome the lack of compactness, and a more careful analysis is needed to consider the relationship between the mountain pass level for Φ and the least energy of the functional associated “limit problem” of (1.1) which is used to recover the compactness; see Lemma 3.4. By using Theorem 1.1 and applying the global compactness lemma and (V1) and (V3) (or (V3′)), we can prove Theorems 1.2 and 1.3; see Lemma 3.5 and Lemma 3.6.

## Proof of Theorem 1.1

First, by a simple calculation, we establish some key inequalities.

### Lemma 2.1

Assume that (F1) and (F6) hold, Then

\begin{aligned} \frac{1}{t^{3}}F\bigl(t^{\beta }\tau \bigr)-F(\tau )+ \frac{1-t^{2\beta -1}}{2 \beta -1}\bigl[\beta f(\tau )\tau -3F(\tau )\bigr] \geq 0, \quad \forall t\geq 0, \tau \in \mathbb{R}. \end{aligned}
(2.1)

### Proof

It is evident that (2.1) holds for $$\tau =0$$. For $$\tau \neq 0$$, let

$$g(t)=\frac{1}{t^{3}}F\bigl(t^{\beta }\tau \bigr)-F(\tau )+\frac{1-t^{2\beta -1}}{2 \beta -1}\bigl[\beta f(\tau )\tau -3F(\tau )\bigr] \geq 0.$$
(2.2)

Then from (F4), one has

\begin{aligned} {g}'(t) =&-\frac{3}{t^{4}}F \bigl(t^{\beta }\tau \bigr)+\frac{\beta }{t^{4}} {f\bigl(t^{\beta }\tau \bigr)}t^{\beta }\tau -t^{2\beta -2}\bigl[\beta f(\tau )\tau -3F(\tau ) \bigr] \\ =&t^{2\beta -2} \biggl[\frac{\beta f(t^{\beta }\tau )t^{\beta }\tau -3F(t ^{\beta }\tau )}{t^{2\beta +2}}-\beta f(\tau )\tau +3F(\tau ) \biggr] \\ & \textstyle\begin{cases} \geq 0, & t\geq 1, \\ \leq 0, & 0< t < 1. \end{cases}\displaystyle \end{aligned}
(2.3)

It follows that $$g(t)\geq g(1)=0$$ for $$t \geq 0$$. This, together with (2.2) implies (2.1) holds. □

Define

\begin{aligned} \kappa (x,t) = &2V(x)-(2\beta -1)t^{2\beta -3}V \bigl(t^{-1}x\bigr)+(2\beta -3)t ^{2\beta -1}V(x) \\ &{}+\bigl[1-t^{2\beta -1}\nabla V(x)\cdot x\bigr], \quad \forall x\in \mathbb{R}^{3}, \forall t>0. \end{aligned}
(2.4)

It is easy to check that (V2) implies

\begin{aligned} \kappa (x,t)>0, \quad \forall x\in \mathbb{R}^{3}, \forall t\in (0,1)\cup (1,\infty ). \end{aligned}
(2.5)

### Lemma 2.2

Assume that (V1), (V4), (F1), (F2) and (F6) hold, Then

\begin{aligned} \begin{aligned}[b] &\varPhi (u) \geq \varPhi \bigl(t^{\beta }u_{t}\bigr)+ \frac{1-t^{2\beta -1}}{2\beta -1}J(u)+ \frac{1}{2(2\beta -1)} \int _{\mathbb{R}^{3}}\kappa (x,t)u^{2}, \\ &\quad \forall u\in H^{1}\bigl(\mathbb{R}^{3}\bigr),\forall t>0. \end{aligned} \end{aligned}
(2.6)

### Proof

Note that

\begin{aligned}[b] \varPhi \bigl(t^{\beta }u_{t} \bigr)= {}&\frac{t^{2\beta -1}}{2} \Vert \nabla u \Vert _{2}^{2}+ \frac{t ^{2\beta -3}}{2} \int _{\mathbb{R}^{3}}V\bigl(t^{-1}x\bigr)u^{2}\,dx\\ &{}+ \frac{t^{2 \beta -1}}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}\,dx -\frac{1}{t^{3}} \int _{\mathbb{R}^{3}}F\bigl(t^{\beta }u\bigr)\,dx. \end{aligned}
(2.7)

Thus, by (1.4), (1.8), (2.1), (2.5) and (2.7), one has

\begin{aligned} &\varPhi (u)-\varPhi \bigl(t^{\beta }u_{t}\bigr) \\ &\quad =\frac{1-t^{2\beta -1}}{2} \Vert \nabla u \Vert _{2}^{2}+ \frac{1}{2} \int _{\mathbb{R}^{3}}\bigl[V(x)-t^{2\beta -3}V\bigl(t^{-1}x \bigr)\bigr]u^{2}(x)\,dx \\ &\qquad {}+\frac{1-t^{2\beta -1}}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}\,dx + \int _{\mathbb{R}^{3}}\bigl[t^{-3}F\bigl(t^{\beta }u \bigr)-F(u)\bigr]\,dx \\ &\quad =\frac{1-t^{2\beta -1}}{2\beta -1} \biggl\{ \frac{2\beta -1}{2} \Vert \nabla u \Vert _{2}^{2}+\frac{1}{2} \int _{\mathbb{R}^{3}}\bigl[(2\beta -3)V(x)- \nabla V(x)\cdot x \bigr]u^{2}\,dx \biggr\} \\ &\qquad {}+\frac{1-t^{2\beta -1}}{2\beta -1} \biggl\{ \frac{2\beta -1}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}- \int _{\mathbb{R}^{3}}\bigl[\beta f(u)u-3F (u)\bigr] \biggr\} \\ &\qquad {}+ \int _{\mathbb{R}^{3}} \biggl\{ \frac{1}{t^{3}}F\bigl(t^{\beta }u \bigr)-F(u)+\frac{1-t ^{2\beta -1}}{2\beta -1}\bigl[\beta f(u)u-3F(u)\bigr] \biggr\} \,dx \\ &\qquad {}+\frac{1}{2(2\beta -1)} \int _{\mathbb{R}^{3}}\kappa (x,t)u^{2}\,dx \\ &\quad \geq \frac{1-t^{2\beta -1}}{2\beta -1}J(u)+\frac{1}{2(2\beta -1)} \int _{\mathbb{R}^{3}}\kappa (x,t)u^{2}\,dx. \end{aligned}
(2.8)

This shows that (2.6) holds. □

Remark that (2.6) with $$t\rightarrow 0$$ implies

\begin{aligned} \varPhi (u)\geq \frac{1}{2\beta -1}J(u)+\frac{1}{2(2\beta -1)} \int _{\mathbb{R}^{3}}\bigl[2V(x)+\nabla V(x)\cdot x\bigr]u^{2} \,dx,\quad \forall u \in H^{1}\bigl(\mathbb{R}^{3}\bigr). \end{aligned}
(2.9)

To overcome the lack of compactness of Sobolev space embedding in $$\mathbb{R}^{3}$$, we define the following energy functional:

\begin{aligned} \begin{aligned}[b] \varPhi ^{\infty }(u)={} & \frac{1}{2} \int _{\mathbb{R}^{3}}\bigl( \vert \nabla u \vert ^{2}+V _{\infty }u^{2}\bigr)\,dx+\frac{1}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha } \ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}\,dx \\ &{}- \int _{\mathbb{R}^{3}}F(u)\,dx. \end{aligned} \end{aligned}
(2.10)

Corresponding to (1.6) and (1.8), we define

\begin{aligned} \mathcal{M}^{\infty }:=\bigl\{ u\in H^{1}\bigl( \mathbb{R}^{3}\bigr)\setminus \{0\}:J ^{\infty }(u)=0\bigr\} \end{aligned}
(2.11)

and

\begin{aligned} \begin{aligned}[b] J^{\infty }(u):= {}& \frac{2\beta -1}{2} \Vert \nabla u \Vert _{2}^{2}+ \frac{2 \beta -3}{2}V_{\infty } \Vert u \Vert _{2}^{2}+ \frac{2\beta -1}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}\,dx \\ &{}- \int _{\mathbb{R}^{3}}\bigl[\beta f(u)u-3F(u)\bigr]\,dx. \end{aligned} \end{aligned}
(2.12)

From Lemma 2.2, we have the following two corollaries.

### Corollary 2.3

Assume that (F1), (F2) and (F6) hold. Then

\begin{aligned}[b] &\varPhi ^{\infty }(u) \geq \varPhi ^{\infty }\bigl(t^{\beta }u_{t}\bigr) + \frac{1-t ^{2\beta -1}}{2\beta -1}J^{\infty }(u) \\ &\hphantom{\varPhi ^{\infty }(u) \geq}{}+\frac{(2\beta -1)(1-t^{2\beta -3})-(2\beta -3)(1-t^{2\beta -1})}{2(2 \beta -1)}V_{\infty } \Vert u \Vert _{2}^{2}, \\ &\quad \forall u\in H^{1}\bigl(\mathbb{R}^{3}\bigr),\forall t\geq 0. \end{aligned}
(2.13)

### Corollary 2.4

Assume that (V1), (V2), (F1) and (F6) hold. Then for $$u\in \mathcal{M}$$

$$\varPhi (u)=\max_{t>0} \varPhi \bigl(t^{\beta }u_{t}\bigr) .$$
(2.14)

From [23, Proposition 2.7], we can obtain the following lemma.

### Lemma 2.5

Assume that (V1) and (V2) hold. Then there exist two constants $$\rho _{1}$$, $$\rho _{2} > 0$$ such that

$$2V(x)+\nabla V(x)\cdot x\geq \rho _{1}$$
(2.15)

and

$$(2\beta -3)V(x)-\nabla V(x)\cdot x\geq \rho _{2}.$$
(2.16)

### Lemma 2.6

Assume that (V1), (V2), (F1), (F2) and (F6) hold. Then, for any $$u\in H^{1}(\mathbb{R}^{3})\setminus \{0\}$$, there exists a unique $$t_{u}>0$$ such that $$t_{u}^{\beta }u_{t_{u}}\in \mathcal{M}$$.

### Proof

Let $$u\in H^{1}(\mathbb{R}^{3})\setminus \{0\}$$ be fixed and define a function $$\zeta (t):=\varPhi (t^{\beta }u_{t})$$ on $$(0,\infty )$$. Clearly, by (1.8) and (2.5), we have

\begin{aligned}[b] \zeta '(t)=0 \quad \Leftrightarrow \quad &\frac{2\beta -1}{2}t^{2\beta -2} \Vert \nabla u \Vert _{2}^{2} \\ &\qquad {}+\frac{t^{2\beta -4}}{2} \int _{\mathbb{R}^{3}}\bigl[(2\beta -3)V\bigl(t^{-1}x\bigr)- \nabla V\bigl(t^{-1}x\bigr)\cdot \bigl(t^{-1}x\bigr) \bigr]u^{2}\,dx \\ &\qquad {} +\frac{2\beta -1}{2q}t^{2\beta -2} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}\,dx \\ &\qquad {}- \int _{\mathbb{R}^{3}}\bigl[\beta f\bigl(t^{\beta }u\bigr)ut ^{\beta -4}-3F\bigl(t^{\beta }u\bigr)t^{-4}\bigr]\,dx \\ &\quad =0 \quad \Leftrightarrow \quad J\bigl(t^{\beta }u_{t} \bigr)=0\quad \Leftrightarrow\quad t^{\beta }u _{t} \in \mathcal{M}. \end{aligned}
(2.17)

By (V1), (F1) and (F5), we have $$\lim_{t\rightarrow 0^{+}}\zeta '(t)=0$$, $$\zeta '(t)>0$$ for $$t>0$$ small and $$\zeta '(t)<0$$ for t large. Therefore $$\max_{t\in [0,+\infty )}\zeta (t)$$ is obtained at $$t_{u}>0$$ so that $$\zeta '(t_{u})=0$$ and $$t_{u}^{\beta }u_{t_{u}}\in \mathcal{M}$$.

Next we claim that $$t_{u}$$ is unique for any $$u\in H^{1}(\mathbb{R} ^{3})\setminus \{0\}$$. In fact, for any given $$u\in H^{1}(\mathbb{R} ^{3})\setminus \{0\}$$, let $$t_{1},t_{2}>0$$ such that $$t_{1}^{\beta }u _{t_{1}},t_{2}^{\beta }u_{t_{2}}\in \mathcal{M}$$. Then $$J(t_{1}^{ \beta }u_{t_{1}})=J(t_{2}^{\beta }u_{t_{2}})=0$$. Jointly with (2.9), we have

\begin{aligned}& \begin{aligned}[b] \varPhi \bigl(t_{1}^{\beta }u_{t_{1}} \bigr)\geq{}& \varPhi \bigl(t_{2}^{\beta }u_{t_{2}}\bigr)\\ &{}+ \frac{1}{2 \beta -1} \frac{t_{2}^{2\beta -1}-t_{1}^{2\beta -1}}{t_{2}^{2\beta -1}}J\bigl(t_{1} ^{\beta }u_{t_{1}} \bigr)+\frac{1}{2(2\beta -1)} \int _{\mathbb{R}^{3}}\kappa \biggl(x,\frac{t_{2}}{t_{1}} \biggr)u^{2}\,dx \\ \geq{}& \varPhi \bigl(t_{2}^{\beta }u_{t_{2}}\bigr)+ \frac{1}{2(2\beta -1)} \int _{\mathbb{R}^{3}}\kappa \biggl(x,\frac{t_{2}}{t_{1}} \biggr)u^{2}\,dx, \end{aligned} \end{aligned}
(2.18)
\begin{aligned}& \begin{aligned}[b] \varPhi \bigl(t_{2}^{\beta }u_{t_{2}} \bigr) &\geq \varPhi \bigl(t_{1}^{\beta }u_{t_{1}}\bigr)+ \frac{1}{2 \beta -1} \frac{t_{1}^{2\beta -1}-t_{2}^{2\beta -1}}{t_{1}^{2\beta -1}}J\bigl(t_{2} ^{\beta }u_{t_{2}} \bigr)\\ &{}+\frac{1}{2(2\beta -1)} \int _{\mathbb{R}^{3}}\kappa \biggl(x,\frac{t_{1}}{t_{2}} \biggr)u^{2}\,dx \\ &\geq \varPhi \bigl(t_{1}^{\beta }u_{t_{1}}\bigr)+ \frac{1}{2(2\beta -1)} \int _{\mathbb{R}^{3}}\kappa \biggl(x,\frac{t_{1}}{t_{2}} \biggr)u^{2}\,dx. \end{aligned} \end{aligned}
(2.19)

Combining (2.18) and (2.19), we have $$t_{1}=t_{2}$$. Therefore, $$t_{u}>0$$ is unique for any $$u\in H^{1}(\mathbb{R}^{3})\setminus \{0 \}$$. □

### Corollary 2.7

Assume that (F1), (F5) and (F6) hold. Then, for any $$u\in H^{1}(\mathbb{R}^{3})\setminus \{0\}$$, there exists a unique $$t_{u}>0$$ such that $$t_{u}^{\beta }u_{t_{u}}\in \mathcal{M}^{\infty }$$.

Combining Corollary 2.4 with Lemma 2.6, we have the following lemma.

### Lemma 2.8

Assume that (V1), (V2), (F1), (F2), (F5) and (F6) hold. Then

$$\inf_{u\in \mathcal{M}}\varPhi (u) =m= \inf_{u\in H^{1}(\mathbb{R}^{3})\setminus \{0\}}\max _{t> 0}\varPhi \bigl(t ^{\beta }u_{t}\bigr).$$

### Lemma 2.9

Assume that (F1) and (F2) hold. If $$u_{n}\rightharpoonup \bar{u}$$ in $$H^{1}(\mathbb{R}^{3})$$, then along a subsequence of $$\{u_{n}\}$$

$$\lim_{n\rightarrow \infty } \sup_{\varphi \in H^{1}(\mathbb{R}^{3}), \Vert \varphi \Vert \leq 1} \biggl\vert \int _{\mathbb{R}^{3}}\bigl[f(u_{n})-f(u_{n}- \bar{u})-f(\bar{u})\bigr]\varphi \,dx \biggr\vert =0.$$
(2.20)

From [33, Proposition 4.3], we can obtain the following Lemma.

### Lemma 2.10

Let $$\alpha \in (0,3)$$ and $$q \in [1,3+\alpha )$$ hold. If $$u_{n} \rightharpoonup \bar{u}$$ in $$H^{1}(\mathbb{R}^{3})$$, then along a subsequence of $$\{u_{n}\}$$

$$\lim_{n\rightarrow \infty } \int _{\mathbb{R}^{3}} \bigl\vert \bigl(I_{\alpha } \ast \vert u_{n} \vert ^{q}\bigr) \vert u_{n} \vert ^{q}-\bigl(I_{\alpha }\ast \vert u_{n}-u \vert ^{q}\bigr) \vert u_{n}-u \vert ^{q}-\bigl(I _{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q} \bigr\vert =0.$$
(2.21)

### Lemma 2.11

Assume that (V1), (V2), (F1), (F5) and (F6) hold. Then

1. (i)

there exists $$\rho > 0$$ such that $$\|u\| \geq \rho$$, $$\forall u\in \mathcal{M}$$;

2. (ii)

$$m=\inf_{\mathcal{M}} \varPhi >0$$.

### Proof

(i) Since $$J(u)=0$$, $$\forall u\in \mathcal{M}$$, by (F1), (F2), (1.8), (2.16) and due to the Sobolev embedding theorem, one has

\begin{aligned}[b] \frac{\min \{2\beta -1,\rho _{2}\}}{2} \Vert u \Vert ^{2} \leq {}& \frac{2\beta -1}{2} \Vert \nabla u \Vert _{2}^{2}+\frac{1}{2} \int _{\mathbb{R} ^{3}}\bigl[(2\beta -3)V(x)-\nabla V(x) \cdot x \bigr]u^{2}\,dx \\ &{}+\frac{2\beta -1}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}\,dx \\ = {}& \int _{\mathbb{R}^{3}}\bigl[\beta f(u)u-3F(u)\bigr]\,dx \\ \leq {}& \frac{\min \{2\beta -1,\rho _{2}\}}{4} \Vert u \Vert ^{2}+ {C_{1}} \Vert u \Vert ^{p}, \end{aligned}
(2.22)

which implies

$$\Vert u \Vert \geq \rho :=\biggl(\frac{\min \{2\beta -1,\rho _{2}\}}{4 {C_{1}}} \biggr)^{\frac{1}{p-2}},\quad \forall u\in \mathcal{M}.$$
(2.23)

(ii) Let $$\{u_{n}\} \subset \mathcal{M}$$ be such that $$\varPhi (u_{n}) \rightarrow m$$. There are two possible cases:

1. (1)

$$\inf_{n\in \mathbb{N}}\|u_{n}\|_{2}>0$$ and

2. (2)

$$\inf_{n\in \mathbb{N}}\|u_{n}\| _{2}=0$$.

Case (1) $$\inf_{n\in \mathbb{N}}\|u_{n}\|_{2}:=\varrho _{1}>0$$. In this case, by (2.9) and (2.15), one has

$$m+o(1)=\varPhi (u_{n})=\varPhi (u_{n})- \frac{1}{2\beta -1}J(u_{n})\geq \frac{ {\rho _{1}}}{2(2\beta -1)}\varrho _{1}^{2}.$$
(2.24)

Case (2) $$\inf_{n\in \mathbb{N}}\|u_{n}\|_{2}:=0$$, by (2.23), passing to a subsequence, we have

$$\Vert u_{n} \Vert _{2}\rightarrow 0, \qquad \Vert \nabla u_{n} \Vert _{2}\geq \frac{1}{2}\rho .$$
(2.25)

Note that (F1) implies that, for any $$\varepsilon >0$$, there exists $$C_{\varepsilon }>0$$ such that

$$\bigl\vert F(t) \bigr\vert \leq C_{\varepsilon } \vert t \vert ^{2}+\varepsilon \vert t \vert ^{6}, \quad \forall t\in \mathbb{R}.$$
(2.26)

By (2.26) and the Sobolev embedding inequality, we have

$$\int _{\mathbb{R}^{3}}F(u)\,dx\leq C_{2} \Vert u \Vert _{2}^{2}+\frac{1}{2}S^{3} \Vert u \Vert _{6}^{6}\leq C_{2} \Vert u \Vert _{2}^{2}+\frac{1}{4} \Vert \nabla u \Vert _{2}^{6}.$$
(2.27)

Let $$t_{n}=\|\nabla u_{n}\|_{2}^{-\frac{2}{2\beta -1}}$$, then (2.25) implies that $$\{t_{n}\}$$ is bounded. Since $$J(u_{n})=0$$, it follows from (2.6), (2.7), (2.25) and (2.27) that

\begin{aligned} m+o(1) =&\varPhi (u_{n})\geq \varPhi \bigl(t_{n}^{\beta }(u_{n})_{t_{n}}\bigr) \\ =&\frac{t_{n}^{2\beta -1}}{2} \Vert \nabla u_{n} \Vert _{2}^{2}+\frac{t_{n} ^{2\beta -3}}{2} \int _{\mathbb{R}^{3}}V\bigl(t^{-1}x\bigr)u_{n}^{2} \,dx \\ &{}+\frac{t _{n}^{2\beta -1}}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u_{n} \vert ^{q}\bigr) \vert u _{n} \vert ^{q}\,dx \\ &{}-t_{n}^{3} \int _{\mathbb{R}^{3}}F\bigl(t_{n}^{\beta }u_{n} \bigr)\,dx \\ \geq &\frac{t_{n}^{2\beta -1}}{2} \Vert \nabla u \Vert _{2}^{2}- \frac{t_{n} ^{6\beta -3}}{4}C_{2} \Vert \nabla u \Vert _{2}^{6} \\ =&\frac{t_{n}^{2\beta -1}}{4} \Vert \nabla u \Vert _{2}^{2} \bigl[2-\bigl(t_{n}^{2 \beta -1} \Vert \nabla u \Vert _{2}^{2}\bigr)^{2} \bigr]=\frac{1}{4}+o(1). \end{aligned}
(2.28)

Case (1) and Case (2) show that $$m=\inf_{\mathcal{M}}\varPhi >0$$. □

### Lemma 2.12

Assume that (V1), (F1) and (F2) hold. If $$u_{n} \rightharpoonup \bar{u}$$ in $$H^{1}(\mathbb{R}^{3})$$, then along a subsequence

$$\begin{gathered} \varPhi (u_{n}) =\varPhi ( \bar{u})+\varPhi (u_{n}-\bar{u})+o(1), \quad J(u_{n}) =J( \bar{u})+J(u_{n}-\bar{u})+o(1), \quad \\ \varPhi '(u_{n}) =\varPhi '(\bar{u})+ \varPhi '(u_{n}-\bar{u})+o(1), \\ \bigl\langle \varPhi '(u_{n}),u_{n}\bigr\rangle =\bigl\langle \varPhi '(\bar{u}),\bar{u} \bigr\rangle +\bigl\langle \varPhi '(u_{n}-\bar{u}),(u_{n}- \bar{u})\bigr\rangle +o(1). \end{gathered}$$
(2.29)

### Lemma 2.13

Assume that (V1), (V2), (F1), (F2), (F5) and (F6) hold. Then $$m^{\infty }:=\inf_{\mathcal{M}^{\infty }}\varPhi ^{\infty } \geq m$$.

### Proof

In view of Lemma 2.2 and Corollary 2.4, we have $$\mathcal{M}^{\infty }\neq \emptyset$$. Arguing indirectly, we assume that $$m>m^{\infty }$$. Let $$\varepsilon :=m-m^{\infty }$$, then there exists $$u_{\varepsilon }^{\infty }$$ such that

\begin{aligned} u_{\varepsilon }^{\infty }\in \mathcal{M}^{\infty } \quad \mbox{and} \quad m^{\infty }+ \frac{\varepsilon }{2}>\varPhi ^{\infty }\bigl(u_{\varepsilon } ^{\infty } \bigr) \end{aligned} .
(2.30)

In view of Corollary 2.7, there exists $$t_{\varepsilon }>0$$ such that $$t_{\varepsilon }^{\beta }(u_{\varepsilon }^{\infty })_{t_{\varepsilon }}\in \mathcal{M}$$. Thus, it follows from (V1), (1.4), (2.10), (2.13) and (2.30) that

\begin{aligned} m^{\infty }+\frac{\varepsilon }{2}> \varPhi ^{\infty }\bigl(u_{\varepsilon } ^{\infty }\bigr)\geq \varPhi ^{\infty }\bigl(t_{\varepsilon }^{\beta }\bigl(u_{\varepsilon }^{\infty } \bigr)_{t_{\varepsilon }}\bigr)\geq \varPhi \bigl(t_{\varepsilon }^{\beta } \bigl(u _{\varepsilon }^{\infty }\bigr)_{t_{\varepsilon }}\bigr)\geq m. \end{aligned}
(2.31)

This contradiction shows that $$m^{\infty }>m$$. □

### Lemma 2.14

Assume that (V1), (V2), (F1), (F2), (F5) and (F6) hold. Then m is obtained.

### Proof

In view of Lemma 2.6 and Lemma 2.11, we have $$\mathcal{M}\neq \emptyset$$ and $$m>0$$. Let $$\{u_{n}\}\subset \mathcal{M}$$ be such that $$\varPhi (u_{n})\rightarrow m$$. Since $$J(u_{n})=0$$, if follows from (2.9) and (2.15) that

\begin{aligned}[b] m+o(1) &=\varPhi (u_{n})=\varPhi (u_{n})-\frac{1}{2(2\beta -1)}J(u_{n}) \\ & \geq \frac{1}{2(2\beta -1)} \int _{\mathbb{R}^{3}}\bigl[2V(x)+\nabla V(x) \cdot x\bigr]u_{n}^{2} \,dx \\ &\geq \frac{\rho _{1}}{2(2\beta -1)} \Vert u_{n} \Vert _{2} ^{2}. \end{aligned}
(2.32)

This shows that $${\|u_{n}\|_{2}}$$ is bounded. Next, we prove that $${\|\nabla u_{n}\|_{2}}$$ is also bounded. Arguing by contradiction, suppose that $$\|\nabla u_{n}\|_{2}\rightarrow \infty$$. By (1.3) and the Sobolev embedding inequality, for $$u\in H^{1}(\mathbb{R}^{3})$$ one has

\begin{aligned} \int _{\mathbb{R}^{3}}F(u)\,dx\leq C_{3} \Vert u \Vert _{2}^{2}+ \frac{1}{2(8m)^{2}}S^{3} \Vert u \Vert _{6}^{6}\leq C_{3} \Vert u \Vert _{2}^{2}+ \frac{1}{4(8m)^{2}} \Vert \nabla u \Vert _{2}^{6}. \end{aligned}
(2.33)

Let $$t_{n}=(8m/\|\nabla u_{n}\|_{2}^{2})^{\frac{1}{2\beta -1}}$$. Since $$J(u_{n})=0$$, it follows from (1.4), (2.6) and (2.32) that

\begin{aligned} m+o(1) =&\varPhi (u_{n})\geq \varPhi \bigl(t_{n}^{\beta }(u_{n})_{t_{n}}\bigr) \\ =&\frac{t_{n}^{2\beta -1}}{2} \Vert \nabla u_{n} \Vert _{2}^{2}+\frac{t_{n} ^{2\beta -3}}{2} \int _{\mathbb{R}^{3}}V\bigl(t_{n}^{-1}x \bigr)u_{n}^{2}\,dx \\ &{}+\frac{t_{n}^{2\beta -1}}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha } \ast \vert u_{n} \vert ^{q}\bigr) \vert u_{n} \vert ^{q} \,dx-t_{n}^{-3} \int _{\mathbb{R}^{3}}F\bigl(t_{n} ^{\beta }u_{n} \bigr) \\ \geq& \frac{t_{n}^{2\beta -1}}{2} \Vert \nabla u_{n} \Vert _{2}^{2}-\frac{C _{3} t_{n}^{2\beta }}{t_{n}^{3}} \Vert u_{n} \Vert _{2}^{2}-\frac{1}{4(8m)^{2}}t _{n}^{6\beta -3} \Vert \nabla u_{n} \Vert _{2}^{6} \\ =&\frac{t_{n}^{2\beta -1}}{2} \Vert \nabla u_{n} \Vert _{2}^{2} \biggl[1- \frac{1}{2}\biggl( \frac{t_{n}^{2\beta -1} \Vert \nabla u_{n} \Vert _{2}^{2}}{8m}\biggr)^{2} \biggr]+o(1) \\ =&2m+o(1). \end{aligned}
(2.34)

This contradiction shows that $$\{\|\nabla u_{n}\|_{2}^{2}\}$$ is also bounded, and so $$\{u_{n}\}$$ is bounded in $$H^{1}(\mathbb{R}^{3})$$. Passing to a subsequence, we have $$u_{n}\rightharpoonup \bar{u}$$ in $$H^{1}(\mathbb{R}^{3})$$. Then $$u_{n}\rightarrow \bar{u}$$ in $$L_{\mathrm{loc}}^{s}(\mathbb{R}^{3})$$ for $$2 \leq s <6$$ and $$u_{n}\rightarrow \bar{u}$$ a.e. in $$\mathbb{R}^{3}$$. There are two possible cases: (i) $$\bar{u}=0$$ and (ii) $$\bar{u}\neq 0$$.

Case (i) $$\bar{u}=0$$. i.e. $$u_{n}\rightharpoonup 0$$ in $$H^{1}( \mathbb{R}^{3})$$. Then $$u_{n}\rightarrow 0$$ in $$L_{\mathrm{loc}}^{s}( \mathbb{R}^{3})$$ for $$2 \leq s <2^{*}$$ and $$u_{n}\rightarrow 0$$ a.e. in $$\mathbb{R}^{3}$$. Using (V1) and (V2), it is easy to show that

\begin{aligned} \lim_{n\rightarrow \infty } \int _{\mathbb{R}^{3}}\bigl[V_{\infty }-V(x)\bigr]u _{n}^{2}\,dx = \lim_{n\rightarrow \infty } \int _{\mathbb{R}^{3}} \nabla V(x) \cdot x u_{n}^{2} \,dx =0 \end{aligned}
(2.35)

From (1.4), (1.8), (2.10), (2.12) and (2.35), one can get

\begin{aligned} \varPhi ^{\infty }(u_{n}) \rightarrow m , \qquad J^{\infty }(u_{n})\rightarrow 0. \end{aligned}
(2.36)

Note that (F1) and (F2) imply that, for any $$\varepsilon >0$$, there exists $$C_{\varepsilon }>0$$ such that

\begin{aligned} \bigl\vert F(t) \bigr\vert \leq \varepsilon \vert t \vert ^{2}+C_{\varepsilon } \vert t \vert ^{p}, \quad \forall t\in \mathbb{R}. \end{aligned}
(2.37)

By (1.3), (1.8), (2.10), (2.37) and Lemma 2.11(i), one has

\begin{aligned}[b] \frac{\min \{\rho _{2},3\}}{2}\rho ^{2} \leq {}&\frac{2\beta -1}{2} \Vert \nabla u_{n} \Vert _{2}^{2}+ \int _{\mathbb{R}^{3}} \biggl[\frac{2\beta -3}{2}V(x)- \frac{1}{2} \nabla V(x) \cdot x \biggr]u_{n}^{2}\,dx \\ &{}+\frac{2\beta -1}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u_{n} \vert ^{q}\bigr) \vert u _{n} \vert ^{q}\,dx \\ ={}& \int _{\mathbb{R}^{3}}\bigl[\beta f(u_{n})u_{n}-3F(u) \bigr]\,dx \\ \leq {}&C_{4}\bigl(\varepsilon \vert u_{n} \vert _{2}^{2}+C_{\varepsilon } \vert u_{n} \vert _{p} ^{p}\bigr). \end{aligned}
(2.38)

Using (2.38) and Lion’s concentration compactness principle [34, Lemma 1.21], we can prove that there exists $$\delta >0$$ and $$y_{n}\in \mathbb{R}^{3}$$, such that $$\int _{B_{1}(y_{n})}|u_{n}|^{2}\,dx> \delta$$. Let $$\hat{u}_{n}(x)=u_{n}(x+y_{n})$$. Then we have $$\|\hat{u}_{n}\|=\|u_{n}\|$$ and

\begin{aligned} J^{\infty }(\hat{u}_{n})=o(1), \qquad \varPhi ^{\infty }(\hat{u}_{n})\rightarrow m, \qquad \int _{B_{1}(0)} \vert \hat{u}_{n} \vert ^{2}\,dx >\delta . \end{aligned}
(2.39)

Therefore, there exists $$\hat{u} \in H^{1}(\mathbb{R}^{3})\setminus \{0\}$$ such that, passing to a subsequence,

$$\textstyle\begin{cases} \hat{u}_{n} \rightharpoonup \hat{u}, & \mbox{in } H^{1}(\mathbb{R}^{3}); \\ \hat{u}_{n} \rightarrow \hat{u}, & \mbox{in } L_{loc}^{s}(\mathbb{R}^{3}), \forall \ s\in [1,6); \\ \hat{u}_{n} \rightarrow \hat{u}, &\mbox{a.e. on } \mathbb{R}^{3}. \end{cases}$$
(2.40)

Let $$w_{n}=\hat{u}_{n}-\hat{u}$$. then (2.40) and Lemma 2.12 yield

$$\varPhi ^{\infty }(\hat{u}_{n})=\varPhi ^{\infty }(\hat{u})+\varPhi ^{\infty }(w_{n})+o(1), \qquad J^{\infty }(\hat{u}_{n})=J^{\infty }(\hat{u})+J^{\infty }(w_{n})+o(1).$$
(2.41)

We define the functional $$\varPsi _{\infty }:H^{1}(\mathbb{R}^{3})\rightarrow \mathbb{R}$$ for $$u\in H^{1}(\mathbb{R}^{3})$$ by

\begin{aligned}[b] \varPsi ^{\infty }(u)&= \varPhi ^{\infty }(u)-\frac{1}{2\beta -1}J^{\infty }(u)\\ &=\frac{V _{\infty }}{2\beta -1} \Vert u \Vert _{2}^{2} +\frac{1}{2(2\beta -1)} \int _{\mathbb{R}^{3}}\bigl[2\beta f(u)u-(4\beta +4)F(u)\bigr]\,dx. \end{aligned}
(2.42)

From (2.5), (2.7), (2.39), (2.41) and (2.42), one has

$$\varPsi ^{\infty }(w_{n})=m-\varPsi ^{\infty }(\hat{u})+o(1), \qquad J^{\infty }(w_{n})=-J^{\infty }( \hat{u})+o(1).$$
(2.43)

If there exists a subsequence $$\{w_{n_{i}}\}$$ of $$w_{n}$$ such that $$w_{n_{i}}=0$$, then we have

$$\varPhi ^{\infty }(\hat{u}) =m, \qquad J^{\infty }( \hat{u})=0.$$
(2.44)

Next, we assume that $$w_{n}\neq 0$$, we claim that $$J^{\infty }(w_{n}) \leq 0$$. Otherwise, if $$J^{\infty }(\hat{u})>0$$, then (2.43) implies $$J^{\infty }(w_{n})< 0$$ for large n. In view of Corollary 2.7, there exists $$t_{n}>0$$ such that $$t_{n}^{\beta }(w_{n})_{t_{n}}\in \mathcal{M}^{\infty }$$ for large n. From (2.5), (2.12), (2.13), (2.43) and Lemma 2.13, we obtain

\begin{aligned} m-\varPsi ^{\infty }(\hat{u})+o(1) =&\varPsi ^{\infty }(w_{n}) \\ =&\varPhi ^{\infty }(w_{n})-\frac{1}{2\beta -1}J^{\infty }(w_{n}) \\ \geq& \varPhi ^{\infty }\bigl(t_{n}^{\beta }(w_{n})_{t_{n}} \bigr)-\frac{t^{2\beta -1}}{2\beta -1}J^{\infty }(w_{n}) \\ &{} + \biggl[\frac{(2\beta -1)(1-t^{2\beta -3})-(2\beta -3)(1-t^{2\beta -3})V _{\infty }}{2(2\beta -1)} \biggr] \Vert \nabla w_{n} \Vert _{2}^{2} \\ \geq& m^{\infty }-\frac{t^{2\beta -1}}{2\beta -1}J^{\infty }(w_{n}) \\ & + \biggl[\frac{(2\beta -1)(1-t^{2\beta -3})-(2\beta -3)(1-t^{2\beta -3})V _{\infty }}{2(2\beta -1)} \biggr] \Vert \nabla w_{n} \Vert _{2}^{2} \\ \geq &m, \end{aligned}
(2.45)

which is a contradiction due to $$\varPsi ^{\infty }(\hat{u})> 0$$. This shows that $$J^{\infty }(\hat{u})\leq 0$$. In view of Lemma 2.6, there exists $$t_{\infty }>0$$ such that $$t_{\infty }^{\beta }\hat{u}_{t_{\infty }} \in \mathcal{M}^{\infty }$$. By (2.5), (2.7), (2.36), (2.39), (2.42), Fatou’s lemma and Lemma 2.13, we have

\begin{aligned} m =&\lim_{n\rightarrow \infty } \biggl[\varPhi ^{\infty }( \hat{u}_{n})-\frac{1}{2 \beta -1}J^{\infty }(\hat{u}_{n}) \biggr] \\ \geq& \varPsi ^{\infty }(\hat{u}_{n})\geq \varPsi ^{\infty }(\hat{u}) = \varPhi ^{\infty }(\hat{u})-\frac{1}{2\beta -1}J^{\infty }( \hat{u}) \\ \geq &\varPhi ^{\infty }\bigl(t_{\infty }^{\beta } \hat{u}_{t_{\infty }}\bigr)-\frac{t ^{2\beta -1}}{2\beta -1}J^{\infty }(\hat{u}) \\ &{} + \biggl[\frac{(2\beta -1)(1-t^{2\beta -3})-(2\beta -3)(1-t^{2\beta -3})V _{\infty }}{2(2\beta -1)} \biggr] \Vert \hat{u} \Vert _{2}^{2} \\ \geq& m^{\infty }-\frac{t^{2\beta -1}}{2\beta -1}J^{\infty }(\hat{u})+ \biggl[ \frac{(2\beta -1)(1-t^{2\beta -3})-(2\beta -3)(1-t^{2\beta -3})V _{\infty }}{2(2\beta -1)} \biggr] \Vert \hat{u} \Vert _{2}^{2} \\ \geq& m, \end{aligned}
(2.46)

which implies (2.44) holds also. In view of Lemma 2.6, there exists $$\hat{t}>0$$ such that $$\hat{t}^{\beta }\hat{u}_{\hat{t}} \in \mathcal{M}$$, moreover, it follows from (V1), (1.3), (2.7), (2.44) and Corollary 2.4 that

$$m \leq \varPhi \bigl(\hat{t}^{\beta }\hat{u}_{\hat{t}} \bigr) \leq \varPhi ^{\infty }\bigl( \hat{t}^{\beta } \hat{u}_{\hat{t}}\bigr)\leq \varPhi ^{\infty }(\hat{u})=m.$$
(2.47)

This shows that m is obtained at $$\hat{t}^{\beta }\hat{u}_{\hat{t}} \in \mathcal{M}$$.

Case (ii) $$\bar{u}\neq 0$$. In this case, analogous to the proof of (2.44), by using Φ and J instead of $$\varPhi ^{\infty }$$ and $$J^{\infty }$$, we can deduce that $$\varPhi (\bar{u})=m$$ and $$J(\bar{u})=0$$. □

In the same way as  or , we can obtain the following lemma.

### Lemma 2.15

Assume that (V1), (V2), (F1), (F2), (F5) and (F6) hold. If $$\bar{u}\in \mathcal{M}$$ and $$\varPhi (\bar{u})=m$$, then ū is a critical point of Φ.

### Proof of Theorem 1.1

In view of Lemma 2.14 and 2.15, there exists $$\bar{u}\in \mathcal{M}$$ such that

\begin{aligned} \varPhi (\bar{u})=m=\inf _{u\in H^{1}(\mathbb{R}^{3})\setminus \{0\}} \max_{t>0}\varPhi \bigl(t^{\beta }u_{t} \bigr), \qquad \varPhi '(\bar{u})=0. \end{aligned}
(2.48)

This shows that ū is a ground state solution of (1.1) such that $$\varPhi (\bar{u})=m=\inf_{\mathcal{M}}\varPhi$$. □

## Proofs of Theorem 1.2 and 1.3

Since we are looking for positive solutions to (1.1), without loss of generality, we suppose that $$f(t)=0$$ for $$t<0$$ in this section.

To use the Jeanjean’s monotonicity trick [18, Theorem 1.1], for $$\lambda \in [1/2,1]$$ we introduce two families of $$\mathcal{C}^{1}$$-functions on $$H^{1}(\mathbb{R}^{3})$$ defined by

\begin{aligned} \varPhi _{\lambda }(u)= \frac{1}{2} \int _{\mathbb{R}^{3}}\bigl( \vert \nabla u \vert ^{2}+V(x)u ^{2}\bigr)\,dx+\frac{1}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}\,dx- \lambda \int _{\mathbb{R}^{3}} F(u)\,dx \end{aligned}
(3.1)

and

\begin{aligned} \varPhi _{\lambda }^{\infty }(u)= \frac{1}{2} \int _{\mathbb{R}^{3}}\bigl( \vert \nabla u \vert ^{2}+V_{\infty }u^{2} \bigr)\,dx+\frac{1}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{ \alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}\,dx- \lambda \int _{\mathbb{R}^{3}} F(u)\,dx. \end{aligned}
(3.2)

In the same way as [13, 31], we can obtain the following lemma.

### Lemma 3.1

Assume that (V1), (V3) (or(V3′)), (F1), (F2), (F5) and (F6) hold. Let u be a critical point of $$\varPhi _{\lambda }$$ in $$H^{1}(\mathbb{R}^{3})$$, then we have the following Pohožaev type identity:

\begin{aligned}[b] P_{\lambda }(u):= {}& \frac{1}{2} \Vert \nabla u \Vert _{2}^{2}+ \frac{1}{2} \int _{\mathbb{R}^{3}}\bigl[3V(x)+\nabla V(x)\cdot x\bigr]u^{2} \,dx+\frac{3+\alpha }{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}\,dx \\ &{}- 3\lambda \int _{\mathbb{R}^{3}} F(u)\,dx. \end{aligned}
(3.3)

We set $$J_{\lambda }(u):= \beta \langle \varPhi _{\lambda }^{\prime }(u),u \rangle -P_{\lambda }(u)$$, then, for $$\lambda \in [1/2,1]$$,

\begin{aligned}[b] J_{\lambda }(u) ={} & \frac{2\beta -1}{2} \Vert \nabla u \Vert _{2}^{2} + \frac{1}{2} \int _{\mathbb{R}^{3}}\bigl[(2\beta -3)V(x)-\nabla V(x)\cdot x\bigr]u ^{2}\,dx \\ &{}+\frac{2\beta -1}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}\,dx \\ &{}-\lambda \int _{\mathbb{R}^{3}}\bigl[\beta f(u)u-3F(u)\bigr]\,dx. \end{aligned}
(3.4)

Correspondingly, for $$\lambda \in [1/2,1]$$ we also let

\begin{aligned}[b] J_{\lambda }^{\infty }(u) ={} &\frac{2\beta -1}{2} \Vert \nabla u \Vert _{2}^{2} + \frac{(2 \beta -3)V_{\infty }}{2} \Vert u \Vert _{2}^{2}+ \frac{2\beta -1}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u \vert ^{q}\bigr) \vert u \vert ^{q}\,dx \\ &{}-\lambda \int _{\mathbb{R}^{3}}\bigl[\beta f(u)u-3F(u)\bigr]\,dx. \end{aligned}
(3.5)

Set

\begin{aligned} \mathcal{M}_{\lambda }^{\infty } := \bigl\{ u\in H^{1}\bigl(\mathbb{R}^{3}\bigr) \setminus \{0 \} :J_{\lambda }^{\infty }(u)=0 \bigr\} , \qquad m_{\lambda }^{\infty }:= \inf_{\mathcal{M}_{\lambda }^{\infty }} \varPhi _{\lambda }^{\infty }. \end{aligned}
(3.6)

By Corollary 2.3, we have the following lemma.

### Lemma 3.2

Assume that (F1), (F5) and (F6) hold. Then

\begin{aligned}[b] &\varPhi _{\lambda }^{\infty }(u) \geq \varPhi _{\lambda }^{\infty }\bigl(t^{ \beta }u_{t} \bigr)+\frac{1-t^{2\beta -1}}{2\beta -1}J_{\lambda }^{\infty }(u) \\ &\hphantom{\varPhi _{\lambda }^{\infty }(u) \geq}{}+\frac{(2\beta -1)(1-t^{2\beta -3})-(2\beta -3)(1-t^{2\beta -1})}{2(2 \beta -1)}V_{\infty } \Vert u \Vert _{2}^{2}, \\ &\quad \forall v\in H^{1}\bigl(\mathbb{R} ^{3}\bigr), t> 0. \end{aligned}
(3.7)

Since $$f(t)=0$$ for $$t<0$$, from Theorem 1.1, the strong maximum principle and standard arguments, we can deduce that $$\varPhi _{1}^{\infty }$$ has a minimizer $$u_{1}^{\infty }>0$$ on $$\mathcal{M}_{1}^{\infty }$$, i.e.

\begin{aligned} u_{1}^{\infty }\in \mathcal{M}_{1}^{\infty }, \qquad \bigl(\varPhi _{1}^{\infty }\bigr)'\bigl(u_{1}^{\infty } \bigr)=0 \quad \mbox{and} \quad m_{1}^{\infty } = \varPhi _{1}^{\infty }\bigl(u_{1}^{\infty }\bigr). \end{aligned}
(3.8)

### Lemma 3.3

Under the assumptions of Theorem 1.2 or Theorem 1.3, we have

1. (i)

There exists $$T>0$$ independent of λ such that $$\varPhi _{\lambda }(T^{\beta }(u_{1}^{\infty })_{T})<0$$ for all $$\lambda \in [1/2,1]$$;

2. (ii)

there exists a positive constant $$\kappa _{0}$$ independent of λ such that, for all $$\lambda \in [1/2,1]$$,

$$c_{\lambda }=\inf_{\gamma \in \tau }\max _{t\in [0,1]} \varPhi _{\lambda }\bigl( \gamma (t)\bigr) \geq \kappa _{0} > \max \bigl\{ \varPhi _{\lambda }(0),\varPhi _{\lambda }\bigl(T^{\beta }\bigl(u_{1}^{\infty } \bigr)_{T}\bigr)\bigr\} ,$$
(3.9)

where $$\tau =\{\gamma \in \mathcal{C}([0,1],H^{1}(\mathbb{R}^{3})): \gamma (0)=0,\gamma (1)=T^{\beta }(u_{1}^{\infty })_{T}\}$$;

3. (iii)

$$c_{\lambda }$$ and $$m_{\lambda }^{\infty }$$ are non-increasing on $$\lambda \in [1/2,1]$$.

The proof of Lemma 3.3 is standard, so we omit it.

### Lemma 3.4

Under the assumptions of Theorem 1.2 or Theorem 1.3, there exists $$\bar{\lambda }\in [1/2,1)$$ such that $$c_{\lambda }< m_{\lambda }^{ \infty }$$ for $$\lambda \in (\bar{\lambda },1]$$.

### Proof

It is easy to see that $$\varPhi _{\lambda }(t^{\beta }(u_{1}^{\infty })_{t})$$ is continuous on $$t\in (0,\infty )$$. Hence for any $$\lambda \in [1/2,1)$$, we can choose $$t_{\lambda }\in (0,T)$$ such that $$\varPhi _{\lambda }(t_{\lambda }^{ \beta }((u_{1}^{\infty })_{t_{\lambda }}))$$. Let $$\beta _{0}= \inf_{\lambda \in [1/2,1]}t_{\lambda }$$. If $$\beta _{0}=0$$, then there exists a sequence $$\{\lambda _{n}\}\subset [1/2,1]$$ such that $$\lambda _{n}\rightarrow \lambda _{0} \in [1/2,1]$$ and $$t_{\lambda _{n}} \rightarrow 0$$, and so by (3.1) and Lemma 3.3(iii), one has

\begin{aligned} 0< c_{1}\leq c_{\lambda _{n}} \leq \varPhi _{\lambda _{n}}\bigl(t_{\lambda _{n}} ^{\beta } \bigl(u_{1}^{\infty }\bigr)_{t_{\lambda _{n}}}\bigr)=o(1). \end{aligned}
(3.10)

This contradiction shows $$\beta _{0}>0$$. Thus $$0<\beta _{0}\leq t_{ \lambda }<T$$ for all $$\lambda \in [1/2,1]$$. Let

\begin{aligned} \bar{\lambda } :=\max \biggl\{ \frac{1}{2},1-\frac{\beta _{0}^{2\beta -3}T^{3}\min_{\beta _{0}\leq s\leq T}\int _{\mathbb{R}^{3}}[V _{\infty }-V(sx)] \vert u_{1}^{\infty } \vert ^{2}\,dx}{2\int _{\mathbb{R}^{3}}F(T ^{\beta }u_{1}^{\infty })\,dx} \biggr\} . \end{aligned}
(3.11)

Then $$1/2 \leq \bar{\lambda } <1$$. From (3.1), (3.2), (3.7), (3.11) and Lemma 3.3(iii), we derive

\begin{aligned} m_{\lambda }^{\infty } \geq &m_{1}^{\infty }= \varPhi _{1}^{\infty }\bigl(u _{1}^{\infty }\bigr) \geq \varPhi _{1}^{\infty }\bigl(t_{\lambda }^{\beta } \bigl(u_{1} ^{\infty }\bigr)_{t_{\lambda }}\bigr) \\ =&\varPhi _{\lambda }\bigl(t_{\lambda }^{\beta } \bigl(u_{1}^{\infty }\bigr)_{t_{\lambda }}\bigr)- \frac{(1-\lambda )}{t_{\lambda }^{3}} \int _{\mathbb{R}^{3}}F\bigl(t_{ \lambda }^{\beta } u_{1}^{\infty }\bigr)\,dx + \frac{t_{\lambda }^{2\beta -3}}{2} \int _{\mathbb{R}^{3}}\bigl[V_{\infty }-V(t _{\lambda }x)\bigr] \bigl\vert u_{1}^{\infty } \bigr\vert ^{2}\,dx \\ \geq &c_{\lambda }-\frac{(1-\lambda )}{T^{3}} \int _{\mathbb{R}^{3}}F\bigl(T ^{\beta }u_{1}^{\infty } \bigr)\,dx+\frac{\beta _{0}^{2\beta -3}}{2} \min_{\beta _{0}\leq s \leq T} \int _{\mathbb{R}^{3}}\bigl[V_{\infty }-V(sx)\bigr] \bigl\vert u_{1}^{\infty } \bigr\vert ^{2}\,dx \\ >&c_{\lambda } , \quad \forall \lambda \in (\bar{\lambda },1]. \end{aligned}
(3.12)

□

### Lemma 3.5

Under the assumptions of Theorem 1.2 or Theorem 1.3, for almost every $$\lambda \in (\bar{\lambda },1]$$, there exists $$u_{\lambda }\in H^{1}( \mathbb{R}^{3})\setminus \{0\}$$ such that.

$$\varPhi _{\lambda }'(u_{\lambda })= 0, \qquad \varPhi _{\lambda }(u_{\lambda })=c_{\lambda }.$$
(3.13)

### Proof

In view of the Jeanjean’s monotonicity trick [36, Theorem 1.1] and Lemma 3.3, for almost every $$\lambda \in [1/2,1]$$, there exists a bounded sequence $$\{u_{n}(\lambda )\}\subset H^{1}(\mathbb{R}^{3})$$, for simplicity, we denote it by $$\{u_{n}\}$$ instead of $$\{u_{n}(\lambda ) \}$$ such that

\begin{aligned} \varPhi _{\lambda }(u_{n}) \rightarrow c_{\lambda }, \qquad \bigl\Vert \varPhi _{\lambda }'(u_{n}) \bigr\Vert \rightarrow 0. \end{aligned}
(3.14)

Using Lemma 2.12, we can deduce that there exists $$u_{\lambda }\in H ^{1}(\mathbb{R}^{3})$$, an integer $$l\in \mathbb{N} \cup \{0\}$$, a sequence $$\{y_{n}^{k}\} \subset \mathbb{R}^{3}$$ and $$w^{k}\in H^{1}( \mathbb{R}^{3})$$ for $$1\leq k \leq l$$ such that $$u_{n}\rightharpoonup u_{\lambda }$$ in $$H^{1}(\mathbb{R}^{3})$$, $$\varPhi _{\lambda }^{\prime }(u_{ \lambda })=0$$, $$(\varPhi _{\lambda }^{\infty })^{\prime }(w^{k})=0$$ and $$\varPhi _{\lambda }^{\infty }(w^{k})\geq m_{\lambda }^{\infty }$$ for $$1 \leq k \leq l$$,

\begin{aligned} \Biggl\Vert u_{n}-u_{\lambda }- \sum_{k=1}^{l}w^{k}\bigl(\cdot + y_{n}^{k}\bigr) \Biggr\Vert \rightarrow 0 \quad \mbox{and} \quad \varPhi _{\lambda }(u_{n})\rightarrow \varPhi _{\lambda }(u_{\lambda })+\sum_{k=1}^{l} \varPhi _{\lambda }^{\infty }\bigl(w^{i}\bigr). \end{aligned}
(3.15)

Since $$\varPhi _{\lambda }^{\prime }(u_{\lambda })=0$$, $$J_{\lambda }(u_{\lambda })=0$$. It follows from (3.1) and (3.4) that

\begin{aligned}[b] \varPhi _{\lambda }(u_{\lambda }) ={}& \varPhi _{\lambda }(u_{\lambda })-\frac{1}{2 \beta -1}J_{\lambda }(u_{\lambda }) \\ ={}&\frac{1}{2(2\beta -1)} \int _{\mathbb{R}^{3}}\bigl[2V(x)+\nabla V(x) \cdot x\bigr]u_{\lambda }^{2} \,dx \\ &{}+\frac{\lambda }{2\beta -1} \int _{\mathbb{R}^{3}}\bigl[\beta f(u_{\lambda })u_{\lambda }-(2 \beta +2)F(u_{\lambda })\bigr]\,dx \geq 0. \end{aligned}
(3.16)

If $$l\neq 0$$, then

\begin{aligned} c_{\lambda } =\lim _{n\rightarrow \infty }\varPhi _{\lambda }(u_{n})= \varPhi _{\lambda }(u_{\lambda })+\sum_{i=1}^{l} \varPhi _{\lambda }^{\infty }\bigl(w ^{i}\bigr)\geq m_{\lambda }^{\infty }, \quad \forall \lambda \in (\bar{\lambda },1], \end{aligned}
(3.17)

which is a contradiction by Lemma 3.4. Thus $$l=0$$, and (3.15) implies that $$u_{n}\rightarrow u_{\lambda }$$ in $$H^{1}(\mathbb{R}^{3})$$ and $$\varPhi _{\lambda }(u_{\lambda })=c_{\lambda }$$ for almost every $$\lambda \in (\bar{\lambda },1]$$. □

### Lemma 3.6

Under the assumptions of Theorem 1.2 or Theorem 1.3, there exists $$\bar{u}>0$$ such that

$$\varPhi '(\bar{u})= 0, \qquad \varPhi ( \bar{u})=c_{1}>0.$$
(3.18)

### Proof

Under the assumptions of Theorem 1.2 or 1.3, in view of Lemma 3.5, there exists two sequences of $$\{\lambda _{n}\} \subset (\bar{\lambda },1]$$ and $$\{u_{\lambda _{n}}\} \subset H^{1}(\mathbb{R}^{3})$$, denoted $$\{u_{n}\}$$, such that

\begin{aligned} \lambda _{n}\rightarrow 1, \qquad \varPhi _{\lambda _{n}}'(u_{n})=0, \qquad \varPhi _{\lambda _{n}}(u_{n})=c_{\lambda _{n}}. \end{aligned}
(3.19)

By (3.1), (3.4), (3.19) and Lemma 3.3(iii), one has

\begin{aligned}[b] c_{1/2} \geq c_{\lambda _{n}} ={}&\varPhi _{\lambda _{n}}(u_{n})- \frac{1}{2 \beta -1}J_{\lambda _{n}}(u_{n}) \\ ={}&\frac{1}{2(2\beta -1)} \int _{\mathbb{R}^{3}}\bigl[2V(x)+\nabla V(x) \cdot x\bigr]u_{n}^{2} \,dx \\ &{}+\frac{\lambda _{n}}{2\beta -1} \int _{\mathbb{R}^{3}}\bigl[\beta f(u_{n})u _{n}-(2 \beta +2)F(u_{n})\bigr]\,dx. \end{aligned}
(3.20)

To prove the boundedness of $$\{\|u_{n}\|\}$$, we distinguish two cases: (1) (V3) holds; (2) (V3′) holds.

Case (1) (V3) holds. In this case, by (F6) and (3.20), one has

$$c_{1/2} \geq \frac{\rho _{1}}{2(2\beta -1)} \Vert u_{n} \Vert _{2}^{2},$$
(3.21)

which implies that $$\{\|u_{n}\|_{2}\}$$ is bounded. Next, we prove that $$\{\|\nabla u_{n}\|_{2}\}$$ is also bounded. Arguing by contradiction, suppose that $$\|\nabla u_{n}\|_{2} \rightarrow \infty$$. By (V1), (V3), (3.21) and Lemma 3.3(iii), one has

$$c_{\lambda _{n}} + \int _{\mathbb{R}^{3}}\bigl[(2\beta -2)V_{\infty }-(2 \beta -2)V(x)- \bigl\vert \nabla V(x)\cdot x \bigr\vert \bigr]u_{n}^{2} \leq M_{0},$$
(3.22)

for some constant $$M_{0}$$. Let $$t_{n}=\min \{1,2(M_{0}/\|\nabla u_{n} \|_{2}^{2})^{1/{2\beta -1}}\}$$, then $$t_{n}\rightarrow 0$$. Thus, it follows from (3.1), (3.2), (3.4), (3.5) and (3.22) that

\begin{aligned} \varPhi _{\lambda _{n}}^{\infty }\bigl(t_{n}^{\beta }(u_{n})_{t_{n}} \bigr) \leq& \varPhi _{\lambda _{n}}^{\infty }(u_{n}) - \frac{1-t^{2\beta -1}}{2\beta -1}J_{\lambda _{n}}^{\infty }(u_{n}) \\ =& \varPhi _{\lambda _{n}}(u_{n}) +\frac{1}{2} \int _{\mathbb{R}^{3}}\bigl[V_{\infty }-V(x)\bigr]u_{n}^{2} \,dx \\ &{}- \frac{1-t^{2\beta -1}}{2\beta -1} \biggl\{ J_{\lambda _{n}}(u_{n})+ \frac{1}{2} \int _{\mathbb{R}^{3}}\bigl[(2\beta -3)V_{\infty } \\ &{}-(2\beta -3)V(x)- \bigl\vert \nabla V(x)\cdot x \bigr\vert \bigr]u_{n}^{2} \,dx \biggr\} \\ \leq& c_{\lambda _{n}} + \int _{\mathbb{R}^{3}}\bigl[(2\beta -2)V_{\infty }-(2 \beta -2)V(x)- \bigl\vert \nabla V(x)\cdot x \bigr\vert \bigr]u_{n}^{2} \leq M_{0}. \end{aligned}
(3.23)

Analogous to the proof of (2.34), we can deduce a contradiction by using (3.23). Hence, $$\{u_{n}\}$$ is bounded in $$H^{1}(\mathbb{R} ^{3})$$ under the assumptions of Theorem 1.2.

Case (2) (V3′) holds. In this case, (V3′) and (3.20) imply

\begin{aligned}[b] c_{1/2} &\geq \frac{\lambda _{n}}{2\beta -1} \int _{\mathbb{R}^{3}}\bigl[ \beta f(u_{n})u_{n}-(2 \beta +2)F(u_{n})\bigr]\,dx \\ &\geq C_{5}\biggl(\mu -\frac{2\beta +2}{\beta }\biggr) \int _{\mathbb{R}^{3}}F(u_{n})\,dx. \end{aligned}
(3.24)

Then it follows from (V1), (3.1), and (3.24) that

\begin{aligned}[b] \frac{\gamma _{0}}{2} \Vert u_{n} \Vert ^{2} & \leq \frac{1}{2} \int _{\mathbb{R} ^{3}} \bigl[ \vert \nabla u_{n} \vert ^{2}+V(x)u_{n}^{2} \bigr]\,dx+\frac{1}{2q} \int _{\mathbb{R}^{3}}\bigl(I_{\alpha }\ast \vert u_{n} \vert ^{q}\bigr) \vert u_{n} \vert ^{q}\,dx \\ &=\lambda \int _{\mathbb{R}^{3}} F(u_{n})\,dx \leq C_{6}, \end{aligned}
(3.25)

where $$\gamma _{0}$$ is a positive constant. Hence, $$\{u_{n}\}$$ is bounded in $$H^{1}(\mathbb{R}^{3})$$ under the assumptions of Theorem 1.3. Similar to the proof of Lemma 3.5, there exists $$\bar{u}\in H^{1}(\mathbb{R} ^{3})\setminus \{0\}$$ such that (3.18) holds. Moreover, by the strong maximum principle and a standard argument, we can conclude that $$\bar{u}>0$$. □

Proofs of Theorem 1.2 and Theorem 1.3 . Let

\begin{aligned} K:= \bigl\{ u\in H^{1}\bigl( \mathbb{R}^{3}\bigr)\setminus \{0\}: \varPhi '(u)=0\bigr\} , \qquad \hat{m}:=\inf_{u\in K}\varPhi (u). \end{aligned}
(3.26)

Then Lemma 3.6 shows that $$K \neq \emptyset$$ and $$\hat{m} \leq c_{1}$$. For any $$u\in K$$, (1.8), (3.4) and Lemma 3.1 imply $$J(u)=J_{1}(u)=\beta \langle \varPhi '(u),u\rangle -\mathcal{P}(u)=0$$. As in (3.16), we have $$\varPhi (u)=\varPhi _{1}(u)\geq 0$$ for any $$u\in K$$, and so $$\hat{m}\geq 0$$. Let $$\{u_{n}\} \subset K$$ such that $$\varPhi '(u_{n})=0$$ and $$\varPhi (u_{n})\rightarrow \hat{m}$$. In view of Lemma 3.4, $$\hat{m}\leq c_{1}\leq m_{1}^{\infty }$$. Similar to the proof of Lemma 3.6, we can deduce that there exists $$\hat{u}>0$$ such that $$\varPhi '(\hat{u})=0$$ and $$\varPhi (\hat{u})=\hat{m}$$. This shows that $$\hat{u}\in H^{1}(\mathbb{R}^{3})$$ is a positive ground state solution of (1.1).

## References

1. 1.

Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger–Maxwell equations. Topol. Methods Nonlinear Anal. 11, 283–293 (1998)

2. 2.

Azzollini, A.: Concentration and compactness in nonlinear Schrödinger–Poisson system with a general nonlinearity. J. Differ. Equ. 249, 1746–1763 (2010)

3. 3.

Azzollini, A., Pomponio, A.: Ground state solutions for the nonlinear Schrödinger–Maxwell equations. J. Math. Anal. Appl. 345, 90–108 (2008)

4. 4.

Cerami, G., Vaira, J.: Positive solutions for some non-autonomous Schrödinger–Poisson systems. J. Differ. Equ. 248, 521–543 (2010)

5. 5.

Chen, S., Tang, X.: Improved results for Klein–Gordon–Maxwell systems with general nonlinearity. Discrete Contin. Dyn. Syst., Ser. A 38, 2333–2348 (2018)

6. 6.

Chen, S., Tang, X.: Infinitely many solutions and least energy solutions for Klein–Gordon–Maxwell systems with general superlinear nonlinearity. Comput. Math. Appl. 75, 3358–3366 (2018)

7. 7.

Sun, J., Ma, S.: Ground state solutions for some Schrödinger–Poisson systems with periodic potentials. J. Differ. Equ. 260, 2119–2149 (2016)

8. 8.

Zhao, L., Zhao, F.: On the existence of solutions for the Schrödinger–Poisson equations. J. Math. Anal. Appl. 346, 155–169 (2008)

9. 9.

Ambrosetti, A., Ruiz, D.: Multiple bound states for the Schrödinger–Poisson equation. Commun. Contemp. Math. 10, 1–14 (2008)

10. 10.

D’Avenia, P.: Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations. Adv. Nonlinear Stud. 2, 177–192 (2002)

11. 11.

D’Aprile, T., Wei, J.: On bound states concentrating on spheres for the Maxwell–Schrödinger equation. SIAM J. Math. Anal. 37, 321–342 (2005)

12. 12.

Coclite, G.: A multiplicity result for the nonlinear Schrödinger–Maxwell equations. Commun. Appl. Anal. 7, 417–423 (2003)

13. 13.

D’Aprile, T., Mugnai, D.: Non-existence results for the coupled Klein–Gordon–Maxwell equations. Adv. Nonlinear Stud. 4, 307–332 (2008)

14. 14.

D’Aprile, T., Mugnai, D.: Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations. Proc. R. Soc. Edinb., Sect. A 134, 1–14 (2004)

15. 15.

Kikuchi, H.: On the existence of a solution for elliptic system related to the Maxwell–Schrödinger equations. Nonlinear Anal. 67, 1445–1456 (2007)

16. 16.

Ruiz, D.: The Schrödinger–Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237, 655–674 (2006)

17. 17.

Ruiz, D.: Existence of groundstate for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 15, 141–164 (2005)

18. 18.

Jeanjean, L.: On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on $$\mathbb {R}^{N}$$. Proc. R. Soc. Edinb., Sect. A 129, 787–809 (1999)

19. 19.

Tang, X., Chen, S.: Ground state solutions of Nehari–Pohoz̆aev type for Schrödinger–Poisson problems with general potentials. Discrete Contin. Dyn. Syst. 37, 4973–5002 (2017)

20. 20.

Shen, L.: Ground state solutions for a class of generalized quasilinear Schrödinger–Poisson systems. Bound. Value Probl. 2018, 44 (2018)

21. 21.

Wen, L., Chen, S.: Ground state solutions for asymptotically periodic Schrödinger–Poisson systems involving Hartree-type nonlinearities. Bound. Value Probl. 2018, 110 (2018)

22. 22.

Chen, S., Tang, X.: Nehari type ground state solutions for asymptotically periodic Schrödinger–Poisson systems. Taiwan. J. Math. 21, 363–383 (2017)

23. 23.

Chen, S., Tang, X.: Ground state solutions of Schrödinger–Poisson systems with variable potential and convolution nonlinearity. J. Math. Anal. Appl. 473, 7–111 (2019)

24. 24.

Chen, J., Gao, Z.: Ground state solutions for fractional Schrödinger equation with variable potential and Berestycki–Lions type nonlinearity. Bound. Value Probl. 2019, 148 (2019)

25. 25.

Chen, S., Shi, J., Tang, X.: Ground state solutions of Nehari–Pohozaev type for the planar Schrödinger–Poisson system with general nonlinearity. Discrete Contin. Dyn. Syst., Ser. A 39, 5867–5889 (2019)

26. 26.

Chen, S., Tang, X.: Berestycki–Lions conditions on ground state solutions for a nonlinear Schrödinger equation with variable potentials. Adv. Nonlinear Anal. 9, 496–515 (2020)

27. 27.

Chen, S., Zhang, B., Tang, X.: Existence and concentration of semiclassical ground state solutions for the generalized Chern–Simons–Schrödinger system in $$H^{1}(\mathbb{R} ^{2})$$. Nonlinear Anal. 185, 68–96 (2019)

28. 28.

Tang, X., Chen, S.: Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki–Lions assumptions. Adv. Nonlinear Anal. 9, 413–437 (2020)

29. 29.

Goubet, O., Hamraoui, E.: Blow-up of solutions to cubic nonlinear Schrödinger equations with defect: the radial case. Adv. Nonlinear Anal. 6, 183–197 (2017)

30. 30.

Papageorgiou, N., Radulescu, V., Repovs, D.: Nonlinear Analysis—Theory and Methods. Springer, Berlin (2019)

31. 31.

Moroz, V., Schaftingen, J.: Existence of groundstate for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367, 6557–6579 (2015)

32. 32.

Tang, X., Chen, S.: Ground state solutions of Nehari–Pohožaev type for Kirchhoff-type problems with general potentials. Calc. Var. Partial Differ. Equ. 56, 110–134 (2017)

33. 33.

Moroz, V., Schaftingen, J.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265, 153–184 (2013)

34. 34.

Willem, M.: Minimax theorems. Prog. Nonlinear Differ. Equ. Appl. 50(1), 139–141 (1996)

35. 35.

Chen, S., Tang, X.: Ground state solutions for generalized quasilinear Schrödinger equations with variable potentials and Berestycki–Lions nonlinearities. J. Math. Phys. 59, 081508 (2018)

36. 36.

Jeanjean, L., Tanka, K.: A positive solution for a nonlinear Schrödinger equation on $$\mathbb{R}^{3}$$. Indiana Univ. Math. J. 54, 443–464 (2005)

## Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

### Funding

This work was supported by the NNSF (11701487, 11626202), Hunan Provincial Natural Science Foundation of China (2016JJ6137), Scientific Research Fund of Hunan Provincial Education Department (15B223).

## Author information

All authors read and approved the final manuscript.

Correspondence to Fangfang Liao.

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### Competing interests

The authors declare that there are no competing interests regarding the publication of this article. 