Ground state solutions of Schrödinger–Poisson systems with asymptotically constant potential

Abstract

This paper is focused on the following Schrödinger–Poisson system $\begin{matrix} \{\begin{matrix} - Δ u + V (x) u + ϕ u = f (u), x \in R^{3}, \\ - Δ ϕ = u^{2}, x \in R^{3}, \end{matrix} \end{matrix}$ where $V (x)$ is weakly differentiable and tends asymptotically to a constant and $f \in C (R, R)$ . By introducing some new tricks, we prove that the above system admits a ground state solution under some mild assumptions on V and f. Our results generalize and improve the existing ones in the literature.

Keywords

Schrödinger–Poisson system ground state solution Pohozaev type identity

1. Introduction

In this paper, we are concerned with the following Schrödinger–Poisson system $\begin{matrix} (1.1) & \{\begin{matrix} - Δ u + V (x) u + ϕ u = f (u), x \in R^{3}, \\ - Δ ϕ = u^{2}, x \in R^{3}, \end{matrix} \end{matrix}$ where V and f satisfy

$V \in C (R^{3}, R)$ and ${inf}_{u \in H^{1} (R^{3}) ∖ {0}} \frac{\int_{R^{3}} [| \nabla u |^{2} + V (x) u^{2}] d x}{\int_{R^{3}} u^{2} d x} : = ϱ_{0} > 0$ ;

$V_{\infty} : = {lim}_{| y | \to \infty} V (y) ⩾ V (x)$ ;

$f \in C (R, R)$ , and there exist constants $C_{0} > 0$ and $p \in (2, 6)$ such that $\begin{matrix} | f (t) | ⩽ C_{0} (1 + | t |^{p - 1}), \forall t \in R; \end{matrix}$

$f (t) = o (t)$ as $t \to 0$ .

Systems like (1.1) have been widely investigated because they have a strong physical meaning. It was first introduced in [4] as a model describing solitary waves for the nonlinear stationary Schrödinger equations interacting with the electrostatic field. It also appears in quantum mechanics models (see e.g. [6,7,22]) and in semiconductor theory [5,25,26].

It is well known that (1.1) can be reduced to a nonlinear Schrödinger equation with a nonlocal term. Indeed, as we shall see in Section 2, for any $u \in H^{1} (R^{3})$ , there exists a unique $ϕ_{u} \in D^{1, 2} (R^{3})$ such that $- Δ ϕ = u^{2}$ by using the Lax–Milgram theorem, then inserted into the first equation, gives $\begin{matrix} (1.2) & - Δ u + V (x) u + ϕ_{u} (x) u = f (u) . \end{matrix}$ Moreover, under our assumptions it is standard to see that (1.2) is variational and its solutions are the critical points of the functional defined in $H^{1} (R^{3})$ by $\begin{matrix} (1.3) & I (u) = \frac{1}{2} \int_{R^{3}} (| \nabla u |^{2} + V (x) u^{2}) d x + \frac{1}{4} \int_{R^{3}} ϕ_{u} (x) u^{2} d x - \int_{R^{3}} F (u) d x, \end{matrix}$ where $F (t) = \int_{0}^{t} f (s) d s$ . Clearly, if $u \in H^{1} (R^{3})$ is a critical point of $I$ , then the pair $(u, ϕ_{u})$ is a solution of (1.1). For the sake of simplicity, in many cases we just say $u \in H^{1} (R^{3})$ , instead of $(u, ϕ_{u}) \in H^{1} (R^{3}) \times D^{1, 2} (R^{3})$ , is a weak solution of (1.1).

In recent years, there has been increasing attention to systems like (1.1) on the existence of positive solutions, ground state solutions, multiple solutions and semiclassical states, see e.g. [2,3,8–13,16–19,28,31–34]. The greatest part of the literature focuses on the study of (1.1) with $V (x) \equiv 1$ or $V (x) = \bar{V} (| x |)$ , and f satisfies the following assumptions of Ambrosetti–Rabinowitz type and 4-superlinear

$f (t) t ⩾ 4 F (t) ⩾ 0$ , $\forall t \in R$ ;

${lim}_{| t | \to \infty} \frac{F (t)}{t^{4}} = \infty$ .

In fact, under (AR) and (SF), it is easy to verify the Mountain Pass geometry and the boundedness of (PS) or ${(C)}_{c}$ sequences for $I$ . If (AR) is replaced by the following stronger monotonic assumption:

the function $f (t) / | t |^{3}$ is increasing on $(- \infty, 0) \cup (0, + \infty)$ ,

then

V (x)

is allowed to be any function which satisfies (V1) and (V2), see [1,3].

For the following special form of (1.1) $\begin{matrix} (1.4) & \{\begin{matrix} - Δ u + u + ϕ u = | u |^{p - 2} u, x \in R^{3}, \\ - Δ ϕ = u^{2}, x \in R^{3}, \end{matrix} \end{matrix}$ there are more results on the existence of solutions. For example, in [14] a radial positive solution of (1.4) is found for $4 < p < 6$ . In [15] a related Pohozaev equality is found. With this equality in hand, the authors can prove that there does not exist nontrivial solutions of (1.4) for $p ⩽ 2$ or $p ⩾ 6$ .

In the case $2 < p ⩽ 4$ , it is very difficult to verify the Mountain Pass geometry and the boundedness of (PS) or ${(C)}_{c}$ sequences for the energy functional associated with (1.4). By introducing Nehari-Pohozaev manifold instead the Nehari manifold, Ruiz [27] proved that (1.4) admits a positive radial solution if $3 < p ⩽ 4$ , but does not have any nontrivial solution for $2 < p ⩽ 3$ . Based on Ruiz’ approach in [27] and a concentration-compactness argument on suitable measures, Azzollini and Pomponio [3] obtained a ground solution of Nehari-Pohozaev type under the same assumption $3 < p ⩽ 4$ as in [27].

The approaches used in [3,27] are successful for (1.4), however, they are no longer applicable for the following system with more general nonlinearity $\begin{matrix} (1.5) & \{\begin{matrix} - Δ u + u + ϕ u = f (u), x \in R^{3}, \\ - Δ ϕ = u^{2}, x \in R^{3} . \end{matrix} \end{matrix}$ Even for the case when $f (u) = | u |^{p_{1} - 2} u + | u |^{p_{2} - 2} u$ with $3 < p_{1} < p_{2} ⩽ 4$ , it is difficult to obtain a ground solution of (1.5) by using the approaches in [3,27]. By using Jeanjean’s monotonicity trick [20] to construct a special (PS) sequence, and by using a Pohozaev identity and a global compactness lemma, Sun and Ma [29] proved that (1.5) admits a ground state solution if f satisfies (F1), (F2) and the following assumption of Ambrosetti–Rabinowitz type

there exists $μ > 3$ such that $f (t) t ⩾ μ F (t) > 0$ for $t \in R ∖ {0}$ .

In paper [30], by introducing new trick, Tang and Chen weaken (AR′) to the following assumptions:

${lim}_{| t | \to \infty} \frac{F (t)}{| t |^{3}} = \infty$ ;

$f (t) t ⩾ 3 F (t) ⩾ 0$ , $\forall t \in R$ , and there exist $κ > 3 / 2$ and $C_{1} > 0$ such that $\begin{matrix} \frac{f (t)}{t} > \frac{γ_{0}}{2} \Rightarrow {| \frac{f (t)}{t} |}^{κ} ⩽ C_{1} [f (t) t - 3 F (t)], \end{matrix}$ where $γ_{0}$ is Sobolev imbedding constant such that $γ_{0} ‖ u ‖_{2} ⩽ ‖ u ‖_{H^{1} (R^{3})}$ for $u \in H^{1} (R^{3})$ .

In [33], by using an approximating method and the monotonicity trick [20], Zhao and Zhao proved that the following nonlinear Schrödinger–Poission system $\begin{matrix} (1.6) & \{\begin{matrix} - Δ u + V (x) u + ϕ u = | u |^{p - 2} u, x \in R^{3}, \\ - Δ ϕ = u^{2}, x \in R^{3} \end{matrix} \end{matrix}$ admits a ground state solution if $p \in (3, 6)$ and V satisfies (V1), (V2) and the following assumption:

$V \in C^{1} (R^{3}, R)$ , and satisfies $\nabla V (x) \cdot x \in L^{\infty} (R^{3}) \cup L^{3 / 2} (R^{3})$ , and $\begin{matrix} 2 V (x) + \nabla V (x) \cdot x ⩾ 0 \forall x \in R^{3} . \end{matrix}$

It is worth pointing out that the Ruiz’ result on the “limit problem” associated with (1.6) plays a key role in the arguments in [33]. Later, the above result was generalized by Tang and Chen [30] to (1.1) via introducing some new tricks. Precisely, (V1)–(V3), (F1), (F2), (F3), (TC) and following assumptions are assumed there,

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

Motivated by the above works, in the present paper, we will show the existence of ground state solutions for (1.1) by using some new analytical techniques and introducing some mild assumptions on V and f. Precisely, in addition to (V1)–(V2) and (F1)–(F3), we make use of following conditions:

$V \in C^{1} (R^{3}, R)$ , and there exist $α > 2$ and $θ_{0} \in [0, 1)$ such that $\begin{matrix} 2 (α - 1) V (x) + \nabla V (x) \cdot x + \frac{θ_{0} (α - 2)}{2 | x |^{2}} ⩾ 0, \forall x \in R^{3} ∖ {0}; \end{matrix}$

$V \in C^{1} (R^{3}, R)$ , and there exist $α > 2$ and $θ_{0} \in [0, 1)$ such that $\begin{matrix} 2 V (x) + \nabla V (x) \cdot x + \frac{θ_{0} (α - 2)}{2 | x |^{2}} ⩾ 0, \forall x \in R^{3} ∖ {0}; \end{matrix}$

$V \in C^{1} (R^{3}, R)$ , and $ess {sup}_{x \in R^{3}} [- \nabla V (x) \cdot x] < \infty$ ;

there exist $α > 2$ such that $α f (t) t - 2 (2 α - 1) F (t) + (α - 1) t^{2} ⩾ 0$ , $\forall t \in R$ ;

$f (t) t ⩾ 0$ for all $t \in R$ , and there exists a $θ_{0} \in [0, 1)$ such that the function $\begin{matrix} t^{- 3} [2 f (t) t - 3 F (t) - \frac{θ_{0} V_{\infty}}{2} t^{2}] \end{matrix}$ is nondecreasing on $(- \infty, 0)$ and $(0, \infty)$ ;

$α f (t) t ⩾ 2 (2 α - 1) F (t)$ , $\forall t \in R$ , where α is the same as in (V4);

$α f (t) t - 2 (2 α - 1) F (t) + (α - 2) V (x) ⩾ 0$ , $\forall (x, t) \in R^{3} \times R$ , where α is the same as in (V4′);

there exists $μ ⩾ 4$ such that $f (t) t ⩾ μ F (t)$ , $\forall t \in R$ .

Now, we state our results of this paper.

Theorem 1.1.
Assume that f satisfies (F1)–(F4). Then (1.5) has a ground state solution $u_{0} \in H^{1} (R^{3}) ∖ {0}$ .
Theorem 1.2.
Assume that V and f satisfy (V1), (V2), (V4), (F1)–(F3), (F5) and (F6). Then (1.1) has a ground state solution $u_{0} \in H^{1} (R^{3}) ∖ {0}$ .
Theorem 1.3.
Assume that V and f satisfy (V1), (V2), (V4′), (F1)–(F3), (F5) and (F6′). Then (1.1) has a ground state solution $u_{0} \in H^{1} (R^{3}) ∖ {0}$ .
Theorem 1.4.
Assume that V and f satisfy (V1), (V2), (V5), (F1)–(F3), (F5) and (F7). Then (1.1) has a ground state solution $u_{0} \in H^{1} (R^{3}) ∖ {0}$ .
Remark 1.5.
Obviously, (F4) is weaker than (AR′), whereas (F4) and (TC) are just different. Moreover, (F4) admits $f (t)$ to be sign-changing. There are many functions which satisfy (F4) but not (TC). For example, when $2 < q < 3 < p < 6$ , $f (t) = a | t |^{p - 2} t + b | t |^{q - 2} t$ satisfies (F4) with $a (p - 3) > b (3 - q) > 0$ and $q (q - 1) (3 - q) b < 1$ or $a > 0 > b$ , but it does not satisfy (TC) for all $a, b > 0$ .
Remark 1.6.
Applying Theorems 1.2 and 1.4 to (1.6), when $3 < p < 4$ , we can weaken (V3) to the following assumption:
$V \in C^{1} (R^{3}, R)$ , and $\begin{matrix} 2 (p - 2) V (x) + (4 - p) \nabla V (x) \cdot x + \frac{θ (p - 3)}{| x |^{2}} ⩾ 0, a.e. x \in R^{3} ∖ {0}; \end{matrix}$
when $4 ⩽ p < 6$ , (V3) can be replaced by (V5).
Remark 1.7.
Let $F (t) = \frac{1 - a + t^{2}}{1 + t^{2}} t^{4}$ , $a \in [0, 1]$ . Then $\begin{matrix} f (t) = 4 t^{3} - \frac{2 a t^{3} (2 + t^{2})}{{(1 + t^{2})}^{2}}, f (t) t - 4 F (t) = \frac{2 a t^{6}}{{(1 + t^{2})}^{2}} ⩾ 0 . \end{matrix}$ It is easy to verify that $f (t)$ satisfy (F1)–(F3), (F5) and (F7). Therefore, if $V (x)$ satisfies (V1), (V2) and (V5), then (1.1) has a ground state solution $u_{0} \in H^{1} (R^{3}) ∖ {0}$ via Theorem 1.4. However, $f (t)$ does not satisfy (MN).

The paper is organized as follows. In Section 2, we give some notation and preliminaries, In Section 3, we give the proof of Theorem 1.1. In Section 4, we complete the proof of Theorems 1.2–1.4.

Throughout this paper, we let $u_{t} (x) : = u (t x)$ for $t > 0$ , and denote the norm of $L^{s} (R^{3})$ by $‖ u ‖_{s} = {(\int_{R^{3}} | u |^{s} d x)}^{1 / s}$ for $s ⩾ 2$ , $B_{r} (x) = {y \in R^{3} : | y - x | < r}$ , and positive constants possibly different in different places, by $C_{1}, C_{2}, \dots$ .
2. Preliminaries and lemmas

Hereafter, $H^{1} (R^{3})$ is the usual Sobolev space with the standard scalar product and norm $\begin{matrix} (u, v) = \int_{R^{3}} (\nabla u \nabla v + u v) d x, ‖ u ‖^{2} = \int_{R^{3}} (| \nabla u |^{2} + u^{2}) d x, \end{matrix}$ and $\begin{matrix} D^{1, 2} (R^{3}) = {u \in L^{6} (R^{3}) : \nabla u \in L^{2} (R^{3})} \end{matrix}$ equipped with the norm defined by $\begin{matrix} ‖ u ‖_{D^{1, 2}}^{2} = \int_{R^{3}} | \nabla u |^{2} d x . \end{matrix}$

It is easy to show that (1.1) can be reduced to a single equation (1.2) with a non-local term. Namely, for any $u^{2} \in L_{loc}^{1} (R^{3})$ such that $\begin{matrix} \int_{R^{3}} \int_{R^{3}} \frac{u^{2} (x) u^{2} (y)}{| x - y |} d x d y < \infty, \end{matrix}$ the distributional solution $\begin{matrix} (2.1) & ϕ_{u} (x) = \int_{R^{3}} \frac{u^{2} (y)}{| x - y |} d y = \frac{1}{| x |} * u^{2} \end{matrix}$ of the Poisson equation $\begin{matrix} - Δ ϕ = u^{2}, x \in R^{3} \end{matrix}$ belongs to $D^{1, 2} (R^{3})$ and is the unique weak solution in $D^{1, 2} (R^{3})$ (see e.g. [27] for more details), and $\begin{array}{l} (2.2) & \int_{R^{3}} \nabla ϕ_{u} \nabla v d x = \int_{R^{3}} u^{2} v d x, \forall v \in H^{1} (R^{3}), \\ (2.3) & \int_{R^{3}} \int_{R^{3}} \frac{u^{2} (x) u^{2} (y)}{| x - y |} d x d y = \int_{R^{3}} ϕ_{u} (x) u^{2} d x . \end{array}$ Moreover, $ϕ_{u} (x) > 0$ when $u \neq 0$ . By using Hardy–Littlewood–Sobolev inequality (see [23] or [24, page 98]), we have the following inequality: $\begin{matrix} (2.4) & \int_{R^{3}} \int_{R^{3}} \frac{| u (x) v (y) |}{| x - y |} d x d y ⩽ \frac{8 \sqrt[3]{2}}{3 \sqrt[3]{π}} ‖ u ‖_{6 / 5} ‖ v ‖_{6 / 5}, u, v \in L^{6 / 5} (R^{3}) . \end{matrix}$ Formally, the solutions of (1.1) are then the critical points of the reduced functional (1.3). Indeed, (V1), (V2), (F1), (F2) and (2.4) imply that $I$ is a well-defined of class $C^{1}$ functional, and that $\begin{matrix} (2.5) & ⟨ I^{'} (u), v ⟩ = \int_{R^{3}} (\nabla u \nabla v + V (x) u v) d x + \int_{R^{3}} [ϕ_{u} (x) u - f (u)] v d x \end{matrix}$

Let $\begin{matrix} (2.6) & J^{\infty} (u) = \frac{3}{2} ‖ \nabla u ‖_{2}^{2} + \frac{V_{\infty}}{2} ‖ u ‖_{2}^{2} + \frac{3}{4} \int_{R^{3}} ϕ_{u} (x) u^{2} d x - \int_{R^{3}} [2 f (u) u - 3 F (u)] d x \end{matrix}$ and $\begin{matrix} (2.7) & M^{\infty} : = {u \in H^{1} (R^{3}) ∖ {0} : J^{\infty} (u) = 0} . \end{matrix}$

It is easy to verify that (F5) is equivalent to the following condition: $\begin{array}{l} [\frac{2 f (τ) τ - 3 F (τ)}{| τ |^{3}} - \frac{2 f (t^{2} τ) t^{2} τ - 3 F (t^{2} τ)}{{(t^{2} | τ |)}^{3}}] sign (1 - t) + θ_{0} V_{\infty} \frac{| 1 - t^{2} |}{2 t^{2} | τ |} ⩾ 0, \\ \forall t > 0, τ \neq 0; \end{array}$

In view of Lemma [30, Lemma 2.2], we have the following lemma.

Lemma 2.1.
Assume that (F1), (F2) and (F5) hold. Then $\begin{array}{rcl} I^{\infty} (u) & ⩾ & I^{\infty} (t^{2} u_{t}) + \frac{1 - t^{3}}{3} J^{\infty} (u) \\ (2.8) & + \frac{(1 - θ_{0}) V_{\infty} {(1 - t)}^{2} (2 + t)}{6} ‖ u ‖_{2}^{2}, \forall u \in H^{1} (R^{3}), t ⩾ 0 . \end{array}$
Lemma 2.2 ([30]).

Assume that (F1), (F2) and (F5) hold. Then for any $u \in H^{1} (R^{3}) ∖ {0}$ , there exists a unique $t_{u} > 0$ such that $t_{u}^{2} u_{t_{u}} \in M^{\infty}$ .

Lemma 2.3 ([30]).

Assume that (F1), (F2), (F3) and (F5) hold. Then (1.1) with $V (x) \equiv V_{\infty}$ has a solution $\bar{u} \in H^{1} (R^{3})$ such that $I^{\infty} (\bar{u}) = {inf}_{M^{\infty}} I^{\infty} = {inf}_{u \in H^{1} (R^{3}) ∖ {0}} {max}_{t > 0} I^{\infty} (t^{2} u_{t}) > 0$ .

By a standard argument, we can prove the following lemma.

Lemma 2.4.
Assume that V satisfies (V1), (V2). Then there exist constants $γ_{1}, γ_{2} > 0$ such that $\begin{matrix} γ_{1} ‖ u ‖^{2} ⩽ \int_{R^{3}} [| \nabla u |^{2} + V (x) u^{2}] d x ⩽ γ_{2} ‖ u ‖^{2}, \forall u \in H^{1} (R^{3}) . \end{matrix}$

3. Ground state solutions for (1.5)

In this section, we give the proof of Theorem 1.1.

The following proposition comes from Jeanjean and Toland [21], which is a generalization of Jeanjean’s monotonicity trick [20].

Proposition 3.1 ([21]).

Let X be a Banach space and let $J \subset R^{+}$ be an interval, and $\begin{matrix} Φ_{λ} (u) = A (u) - λ B (u), \forall λ \in J, \end{matrix}$ be a family of $C^{1}$ -functional on X such that

either $A (u) \to + \infty$ or $B (u) \to + \infty$ , as $‖ u ‖ \to \infty$ ;

B maps every bounded set of X into a set of $R$ bounded below;

there are two points $v_{1}$ , $v_{2}$ in X such that $\begin{matrix} (3.1) & c_{λ} : = inf_{γ \in Γ} max_{t \in [0, 1]} Φ_{λ} (γ (t)) > max {Φ_{λ} (v_{1}), Φ_{λ} (v_{2})}, \end{matrix}$

where

\begin{matrix} Γ = {γ \in C ([0, 1], X) : γ (0) = v_{1}, γ (1) = v_{2}} . \end{matrix}

Then, for almost every

λ \in J

, there exists a sequence such that

${u_{n} (λ)}$ is bounded in X;

$Φ_{λ} (u_{n} (λ)) \to c_{λ}$ ;

$Φ_{λ}^{'} (u_{n} (λ)) \to 0$ in $X^{*}$ , where $X^{*}$ is the dual of X.

To apply Proposition 3.1, we use the idea employed by Jeanjean and Tanaka [20] which is an approximation procedure. Precisely, for any $λ \in [1 / 2, 1]$ we study the functional $I_{λ}^{0} : H^{1} (R^{3}) \to R$ defined by $\begin{matrix} (3.2) & I_{λ}^{0} (u) = \frac{1}{2} (‖ \nabla u ‖_{2}^{2} + ‖ u ‖_{2}^{2}) + \frac{1}{4} \int_{R^{3}} ϕ_{u} (x) u^{2} d x - λ \int_{R^{3}} F (u) d x . \end{matrix}$ When $λ = 1$ , (3.2) reduces to $\begin{matrix} (3.3) & I^{0} (u) = \frac{1}{2} \int_{R^{3}} (| \nabla u |^{2} + u^{2}) d x + \frac{1}{4} \int_{R^{3}} ϕ_{u} (x) u^{2} d x - \int_{R^{3}} F (u) d x . \end{matrix}$ Obviously, $I^{0}, I_{λ}^{0} \in C^{1} (H^{1} (R^{3}), R)$ , and the critical points of (3.3) are the solutions of (1.5).

Lemma 3.2 ([29]).

Assume that (F1)–(F3) hold. Let u be a critical point of $I_{λ}^{0}$ in $H^{1} (R^{3})$ , then we have the following Pohozaev type identity $\begin{matrix} (3.4) & P_{λ}^{0} (u) : = \frac{1}{2} (‖ \nabla u ‖_{2}^{2} + 3 ‖ u ‖_{2}^{2}) + \frac{5}{4} \int_{R^{3}} ϕ_{u} (x) u^{2} d x - 3 λ \int_{R^{3}} F (u) d x = 0 . \end{matrix}$

Similarly to the proofs of [29, Lemma 5.2, Lemma 5.3], we can prove the following two lemmas:

Lemma 3.3.
Assume that (F1)–(F3) hold. Then
there exists $v \in H^{1} (R^{3})$ independent of λ such that $I_{λ}^{0} (v) < 0$ for all $λ \in [1 / 2, 1]$ ;

there exists a positive constant $κ_{0}$ independent of λ such that for all $λ \in [1 / 2, 1]$ , $\begin{matrix} (3.5) & c_{λ} : = inf_{γ \in Γ} max_{t \in [0, 1]} I_{λ}^{0} (γ (t)) ⩾ κ_{0} > max {I_{λ}^{0} (0), I_{λ}^{0} (v)}, \end{matrix}$ where $\begin{matrix} Γ = {γ \in C ([0, 1], H^{1} (R^{3})) : γ (0) = 0, γ (1) = v}; \end{matrix}$

there exists a positive constant M independent of λ such that $c_{λ} ⩽ M$ for all $λ \in [1 / 2, 1]$ .

Lemma 3.4.
Assume that (F1)–(F3) hold. Then for almost every $λ \in [1 / 2, 1]$ , there exists $u_{λ} \in H^{1} (R^{3}) ∖ {0}$ such that $\begin{matrix} (3.6) & {(I_{λ}^{0})}^{'} (u_{λ}) = 0, I_{λ}^{0} (u_{λ}) = c_{λ} . \end{matrix}$
Proof of Theorem 1.1.
In view of Lemmas 3.2 and 3.4, there exist two sequences of ${λ_{n}} \subset [1 / 2, 1]$ and ${u_{λ_{n}}} \subset H^{1} (R^{3})$ , denoted by ${u_{n}}$ , such that $\begin{matrix} (3.7) & λ_{n} \to 1, {(I_{λ_{n}}^{0})}^{'} (u_{n}) = 0, P_{λ_{n}}^{0} (u_{n}) = 0, I_{λ_{n}}^{0} (u_{n}) = c_{λ_{n}} . \end{matrix}$ From (F4), (3.2), (3.4), (3.7) and Lemma 3.4(iii), one has $\begin{array}{rcl} M & ⩾ & c_{λ_{n}} = I_{λ_{n}}^{0} (u_{n}) - \frac{1}{4 α - 5} [α ⟨ {(I_{λ_{n}}^{0})}^{'} (u_{n}), u_{n} ⟩ - P_{λ_{n}}^{0} (u_{n})] \\ = & \frac{α - 2}{4 α - 5} ‖ \nabla u_{n} ‖_{2}^{2} + \frac{α - 1}{4 α - 5} ‖ u_{n} ‖_{2}^{2} \\ + \frac{λ_{n}}{4 α - 5} \int_{R^{3}} [α f (u_{n}) u_{n} - 2 (2 α - 1) F (u_{n})] d x \\ ⩾ & \frac{α - 2}{4 α - 5} ‖ \nabla u_{n} ‖_{2}^{2} + \frac{λ_{n}}{4 α - 5} \int_{R^{3}} [α f (u_{n}) u_{n} - 2 (2 α - 1) F (u_{n}) + (α - 1) u_{n}^{2}] d x \\ (3.8) & ⩾ & \frac{α - 2}{4 α - 5} ‖ \nabla u_{n} ‖_{2}^{2} . \end{array}$ This shows that ${‖ \nabla u_{n} ‖_{2}}$ is bounded. Next, we demonstrate that ${u_{n}}$ is bounded in $H^{1} (R^{3})$ . According to (F1), (F2), (3.2), (3.7), (3.8) and the Sobolev embedding theorem, we have $\begin{array}{rcl} ‖ u_{n} ‖_{2}^{2} & ⩽ & \int_{R^{3}} (| \nabla u_{n} |^{2} + u_{n}^{2}) d x \\ ⩽ & 2 c_{λ_{n}} + 2 λ_{n} \int_{R^{3}} F (u_{n}) d x \\ ⩽ & 2 M + \frac{1}{4} ‖ u_{n} ‖_{2}^{2} + C_{1} ‖ u_{n} ‖_{6}^{6} \\ ⩽ & 2 M + \frac{1}{4} ‖ u_{n} ‖_{2}^{2} + C_{1} S^{- 3} ‖ \nabla u_{n} ‖_{2}^{6} . \end{array}$ Hence, ${‖ u_{n} ‖_{2}}$ is bounded, and so ${u_{n}}$ is bounded in $H^{1} (R^{3})$ . In view of Lemma 3.4(iii), we have ${lim}_{n \to \infty} c_{λ_{n}} = c_{} ⩽ M$ . Hence, it follows from (3.2), (3.3), (3.5) and (3.7) that $\begin{matrix} (3.9) & I^{0} (u_{n}) \to c_{} ⩾ κ_{0}, {(I^{0})}^{'} (u_{n}) \to 0 . \end{matrix}$ By a standard argument, we can show that there exists $\tilde{u} \in H^{1} (R^{3}) ∖ {0}$ such that $\begin{matrix} (3.10) & {(I^{0})}^{'} (\tilde{u}) = 0, 0 < I^{0} (\tilde{u}) ⩽ M . \end{matrix}$ Let $\begin{matrix} K : = {u \in H^{1} (R^{3}) ∖ {0} : {(I^{0})}^{'} (u) = 0}, \hat{m} : = inf_{u \in K} I^{0} (u) . \end{matrix}$ Then (3.10) shows that $K \neq \emptyset$ and $\hat{m} ⩽ M$ . For any $u \in K$ , Lemma 3.2 implies $\begin{matrix} (3.11) & P^{0} (u) : = \frac{1}{2} (‖ \nabla u ‖_{2}^{2} + 3 ‖ u ‖_{2}^{2}) + \frac{5}{4} \int_{R^{3}} ϕ_{u} (x) u^{2} d x - 3 \int_{R^{3}} F (u) d x = 0 . \end{matrix}$ By (3.3), (F1), (F2) and the fact $⟨ {(I^{0})}^{'} (u), u ⟩ = 0$ , it is easy to verify that $\begin{matrix} (3.12) & δ : = inf {‖ \nabla u ‖_{2} : u \in K} > 0 . \end{matrix}$ Then we deduce from (F4), (3.3), (3.11) and (3.12) that $\begin{array}{rcl} I^{0} (u) & = & I^{0} (u) - \frac{1}{4 α - 5} [α ⟨ {(I^{0})}^{'} (u), u ⟩ - P^{0} (u)] \\ = & \frac{α - 2}{4 α - 5} ‖ \nabla u ‖_{2}^{2} + \frac{α - 1}{4 α - 5} ‖ u ‖_{2}^{2} \\ + \frac{1}{4 α - 5} \int_{R^{3}} [α f (u) u - 2 (2 α - 1) F (u)] d x \\ ⩾ & \frac{α - 2}{4 α - 5} ‖ \nabla u ‖_{2}^{2} \\ (3.13) & ⩾ & \frac{α - 2}{4 α - 5} δ^{2} . \end{array}$ Hence $\hat{m} > 0$ . Let ${u_{n}} \subset K$ such that $\begin{matrix} (3.14) & {(I^{0})}^{'} (u_{n}) = 0, P^{0} (u_{n}) = 0, I^{0} (u_{n}) \to \hat{m} . \end{matrix}$ By a standard argument, we can prove that there exists $\bar{u} \in H^{1} (R^{N}) ∖ {0}$ such that $u_{n} ⇀ \bar{u}$ in $H^{1} (R^{3})$ , $u_{n} \to \bar{u}$ a.e. on $R^{3}$ and ${(I^{0})}^{'} (\bar{u}) = 0$ . Obviously, $P^{0} (\bar{u}) = 0$ and $I^{0} (\bar{u}) ⩾ \hat{m}$ . On the other hand, from (F4), (3.3), (3.11), (3.14), the weak semi-continuity of norm and Fatou’s Lemma, we have $\begin{array}{rcl} \hat{m} & = & lim_{n \to \infty} {I^{0} (u_{n}) - \frac{1}{4 α - 5} [α ⟨ {(I^{0})}^{'} (u_{n}), u_{n} ⟩ - P^{0} (u_{n})]} \\ = & lim_{n \to \infty} {\frac{α - 2}{4 α - 5} ‖ \nabla u_{n} ‖_{2}^{2} + \frac{α - 1}{4 α - 5} ‖ u_{n} ‖_{2}^{2} \\ + \frac{1}{4 α - 5} \int_{R^{3}} [α f (u_{n}) u_{n} - 2 (2 α - 1) F (u_{n})] d x} \\ = & lim_{n \to \infty} {\frac{α - 2}{4 α - 5} ‖ \nabla u_{n} ‖_{2}^{2} + \frac{1}{4 α - 5} \int_{R^{3}} [α f (u_{n}) u_{n} - 2 (2 α - 1) F (u_{n}) + (α - 1) u_{n}^{2}] d x} \\ ⩾ & \frac{α - 2}{4 α - 5} ‖ \nabla \bar{u} ‖_{2}^{2} + \frac{1}{4 α - 5} \int_{R^{3}} [α f (\bar{u}) \bar{u} - 2 (2 α - 1) F (\bar{u}) + (α - 1) {\bar{u}}^{2}] d x \\ = & I^{0} (\bar{u}) - \frac{1}{4 α - 5} [α ⟨ {(I^{0})}^{'} (\bar{u}), \bar{u} ⟩ - P^{0} (\bar{u})] \\ (3.15) & = & I^{0} (\bar{u}), \end{array}$ which implies that $I^{0} (\bar{u}) ⩽ \hat{m}$ . Therefore, we get $\begin{matrix} (3.16) & {(I^{0})}^{'} (\bar{u}) = 0, I^{0} (\bar{u}) = \hat{m} > 0 . \end{matrix}$ This shows that $\bar{u}$ is a ground state solution of (1.5). □

4. Ground state solutions for (1.1)

In this section, we give the proofs of Theorems 1.2–1.4. To this end, we assume that $V (x) ⩽ V_{\infty}$ but $V (x) ≢ V_{\infty}$ .

Proposition 4.1 ([20]).

Let X be a Banach space and let $J \subset R^{+}$ be an interval. We consider a family ${Φ_{λ}}_{λ \in J}$ of $C^{1}$ -functional on X of the form $\begin{matrix} Φ_{λ} (u) = A (u) - λ B (u), \forall λ \in J, \end{matrix}$ where $B (u) ⩾ 0$ , $\forall u \in X$ , and such that either $A (u) \to + \infty$ or $B (u) \to + \infty$ , as $‖ u ‖ \to \infty$ . We assume that there are two points $v_{1}$ , $v_{2}$ in X such that $\begin{matrix} (4.1) & c_{λ} : = inf_{γ \in Γ} max_{t \in [0, 1]} Φ_{λ} (γ (t)) > max {Φ_{λ} (v_{1}), Φ_{λ} (v_{2})}, \end{matrix}$ where $\begin{matrix} Γ = {γ \in C ([0, 1], X) : γ (0) = v_{1}, γ (1) = v_{2}} . \end{matrix}$ Then, for almost every $λ \in J$ , there is a bounded ${(PS)}_{c_{λ}}$ sequence for $Φ_{λ}$ , that is, there exists a sequence such that

${u_{n} (λ)}$ is bounded in X;

$Φ_{λ} (u_{n} (λ)) \to c_{λ}$ ;

$Φ_{λ}^{'} (u_{n} (λ)) \to 0$ in $X^{*}$ , where $X^{*}$ is the dual of X.

To apply Proposition 4.1, we introduce two families of functional defined by $\begin{matrix} (4.2) & I_{λ} (u) = \frac{1}{2} \int_{R^{3}} (| \nabla u |^{2} + V (x) u^{2}) d x + \frac{1}{4} \int_{R^{3}} ϕ_{u} (x) u^{2} d x - λ \int_{R^{3}} F (u) d x \end{matrix}$ and $\begin{matrix} (4.3) & I_{λ}^{\infty} (u) = \frac{1}{2} \int_{R^{3}} (| \nabla u |^{2} + V_{\infty} u^{2}) d x + \frac{1}{4} \int_{R^{3}} ϕ_{u} (x) u^{2} d x - λ \int_{R^{3}} F (u) d x, \end{matrix}$ for $λ \in [1 / 2, 1]$ .

Lemma 4.2 ([30]).

Assume that $V \in C^{1} (R^{3}, R)$ and (V1), (V2), (F1) and (F2) hold. Let u be a critical point of $I_{λ}$ in $H^{1} (R^{3})$ , then we have the following Pohozaev type identity $\begin{array}{rcl} P_{λ} (u) & : = & \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{1}{2} \int_{R^{3}} [3 V (x) + \nabla V (x) \cdot x] u^{2} d x \\ (4.4) & + \frac{5}{4} \int_{R^{3}} ϕ_{u} (x) u^{2} d x - 3 λ \int_{R^{3}} F (u) d x = 0 . \end{array}$

We set $\begin{matrix} (4.5) & J_{λ}^{\infty} (u) = \frac{3}{2} ‖ \nabla u ‖_{2}^{2} + \frac{V_{\infty}}{2} ‖ u ‖_{2}^{2} + \frac{3}{4} \int_{R^{3}} ϕ_{u} (x) u^{2} d x - λ \int_{R^{3}} [2 f (u) u - 3 F (u)] d x \end{matrix}$ and $\begin{matrix} (4.6) & M_{λ}^{\infty} : = {u \in H^{1} (R^{3}) ∖ {0} : J_{λ}^{\infty} (u) = 0}, m_{λ}^{\infty} : = inf_{u \in M_{λ}^{\infty}} I_{λ}^{\infty} (u), \end{matrix}$ for $λ \in [0.5, 1]$ . In view of Lemma 2.3, under (F1)–(F3) and (F5), for every $λ \in [1 / 2, 1]$ , $I_{λ}^{\infty}$ has a minimizer $u_{λ}^{\infty} \neq 0$ on $M_{λ}^{\infty}$ , i.e. $\begin{matrix} (4.7) & u_{λ}^{\infty} \in M_{λ}^{\infty}, {(I_{λ}^{\infty})}^{'} (u_{λ}^{\infty}) = 0 and m_{λ}^{\infty} = I_{λ}^{\infty} (u_{λ}^{\infty}) . \end{matrix}$

By Lemma 2.1, we have the following lemma.

Lemma 4.3.
Assume that (F1), (F2) and (F5) hold. Then $\begin{array}{rcl} I_{λ}^{\infty} (u) & ⩾ & I_{λ}^{\infty} (t^{2} u_{t}) + \frac{1 - t^{3}}{3} J_{λ}^{\infty} (u) \\ (4.8) & + \frac{(1 - θ_{0}) V_{\infty} {(1 - t)}^{2} (2 + t)}{6} ‖ u ‖_{2}^{2}, \forall u \in H^{1} (R^{3}), t ⩾ 0 . \end{array}$
Lemma 4.4.
Assume that (V1), (V2), (F1)–(F3) and (F5) hold. Then
there exists $T > 0$ independent of λ such that $I_{λ} (T^{2} {(u_{1}^{\infty})}_{T}) < 0$ for all $λ \in [1 / 2, 1]$ ;

there exists a positive constant $κ_{0}$ independent of λ such that for all $λ \in [1 / 2, 1]$ , $\begin{matrix} (4.9) & c_{λ} : = inf_{γ \in Γ} max_{t \in [0, 1]} I_{λ} (γ (t)) ⩾ κ_{0} > max {I_{λ} (0), I_{λ} (T^{2} {(u_{1}^{\infty})}_{T})}, \end{matrix}$ where $\begin{matrix} Γ = {γ \in C ([0, 1], H^{1} (R^{3})) : γ (0) = 0, γ (1) = {(T^{2} u_{1}^{\infty})}_{T}}; \end{matrix}$

$c_{λ}$ is non-increasing on $λ \in [1 / 2, 1]$ ;

$m_{λ}^{\infty}$ is non-increasing on $λ \in [1 / 2, 1]$ ;

${lim sup}_{λ \to λ_{0}} c_{λ} ⩽ c_{λ_{0}}$ for $λ_{0} \in (1 / 2, 1]$ .

Since $m_{λ}^{\infty} = I_{λ}^{\infty} (u_{λ}^{\infty})$ and $\int_{R^{3}} F (u_{λ}^{\infty}) d x > 0$ , then the proofs of (i)–(iv) in Lemma 4.4 are standard, (v) can be proved similarly to [20, Lemma 2.3], so we omit it.

Since (1.1) with $V (x) \equiv V_{\infty}$ is autonomous, $V \in C (R^{3}, R)$ and $V (x) ⩽ V_{\infty}$ but $V (x) ≢ V_{\infty}$ , then there exist $\bar{x} \in R^{3}$ and $\bar{r} \in (0, 1 / 2)$ such that $\begin{matrix} (4.10) & V_{\infty} - V (x) > 0, | u_{1}^{\infty} (x) | > 0 a.e. | x - \bar{x} | ⩽ \bar{r} . \end{matrix}$
Lemma 4.5.
Assume that (V1), (V2), (F1)–(F3) and (F5) hold. Then there exists $\bar{λ} \in [1 / 2, 1)$ such that $c_{λ} < m_{λ}^{\infty}$ for $λ \in (\bar{λ}, 1]$ .
Proof.
It is easy to see that $I_{λ} (t^{2} {(u_{1}^{\infty})}_{t})$ is continuous on $t \in [0, \infty)$ . Hence for any $λ \in [1 / 2, 1)$ , we can choose $t_{λ} \in (0, T)$ such that $I_{λ} (t_{λ}^{2} {(u_{1}^{\infty})}_{t_{λ}}) = {max}_{t \in [0, T]} I_{λ} (t^{2} {(u_{1}^{\infty})}_{t})$ . Note that $I_{1 / 2} (t^{2} {(u_{1}^{\infty})}_{t}) \to - \infty$ as $t \to \infty$ , thus there exists $T_{0} > 0$ such that $\begin{matrix} (4.11) & I_{1 / 2} (t^{2} {(u_{1}^{\infty})}_{t}) ⩽ I_{1} (u_{1}^{\infty}) - 1, \forall t ⩾ T_{0} . \end{matrix}$ Noting that $F (t) ⩾ 0$ . Then it follows from (4.2) and the definition of $t_{λ}$ that $\begin{matrix} I_{1} (u_{1}^{\infty}) ⩽ I_{λ} (u_{1}^{\infty}) ⩽ I_{λ} (t_{λ}^{2} {(u_{1}^{\infty})}_{t_{λ}}) ⩽ I_{1 / 2} (t_{λ}^{2} {(u_{1}^{\infty})}_{t_{λ}}), \forall λ \in [1 / 2, 1], \end{matrix}$ which, together with (4.11), implies $t_{λ} < T_{0}$ for $λ \in [1 / 2, 1]$ . Let $β_{0} = {inf}_{λ \in [1 / 2, 1]} t_{λ}$ . If $β_{0} = 0$ , then there exists a sequence ${λ_{n}} \subset [1 / 2, 1]$ such that $\begin{matrix} λ_{n} \to λ_{0} \in [1 / 2, 1] and t_{λ_{n}} \to 0 . \end{matrix}$ It follows that $\begin{matrix} 0 < c_{1} ⩽ c_{λ_{n}} ⩽ I_{λ_{n}} (t_{λ_{n}}^{2} {(u_{1}^{\infty})}_{t_{λ_{n}}}) = o (1), \end{matrix}$ which implies $β_{0} > 0$ . Thus $\begin{matrix} (4.12) & 0 < β_{0} ⩽ t_{λ} < T_{0}, \forall λ \in [1 / 2, 1] . \end{matrix}$ Let $\begin{matrix} (4.13) & ζ_{0} : = \frac{\bar{r}}{2 (1 + \bar{r} + | \bar{x} |)} . \end{matrix}$ Then it follows from (4.13) that $\begin{matrix} (4.14) & | x - \bar{x} | ⩽ \frac{\bar{r}}{2} and s \in [1 - ζ_{0}, 1 + ζ_{0}] \Rightarrow | s^{- 1} x - \bar{x} | ⩽ \bar{r} . \end{matrix}$

Let $\begin{array}{rcl} \bar{λ} & : = & max {\frac{1}{2}, 1 - \frac{{(1 - ζ_{0})}^{4} {min}_{1 - ζ_{0} ⩽ s ⩽ 1 + ζ_{0}} \int_{R^{3}} [V_{\infty} - V (s^{- 1} x)] | u_{1}^{\infty} |^{2} d x}{2 \int_{R^{3}} F (T_{0}^{2} u_{1}^{\infty}) d x}, \\ (4.15) & 1 - \frac{(1 - θ) V_{\infty} ζ_{0}^{2} β_{0}^{3} ‖ u_{1}^{\infty} ‖_{2}^{2}}{3 \int_{R^{3}} F (T_{0}^{2} u_{1}^{\infty}) d x}} . \end{array}$ Then it follows from (4.10), (4.12), (4.13), (4.14) and (4.15) that $1 / 2 ⩽ \bar{λ} < 1$ . We have two cases to distinguish:

Case i). $t_{λ} \in [1 - ζ_{0}, 1 + ζ_{0}]$ . From (4.2), (4.3), (4.8), (4.12), (4.15) and Lemma 4.4(iii)–(iv), we have $\begin{array}{rcl} m_{λ}^{\infty} & ⩾ & m_{1}^{\infty} = I_{1}^{\infty} (u_{1}^{\infty}) ⩾ I_{1}^{\infty} (t_{λ}^{2} {(u_{1}^{\infty})}_{t_{λ}}) \\ = & I_{λ} (t_{λ}^{2} {(u_{1}^{\infty})}_{t_{λ}}) - \frac{1 - λ}{t_{λ}^{3}} \int_{R^{3}} F (t_{λ}^{2} u_{1}^{\infty}) d x + \frac{t_{λ}}{2} \int_{R^{3}} [V_{\infty} - V (t_{λ}^{- 1} x)] {| u_{1}^{\infty} |}^{2} d x \\ ⩾ & c_{λ} - \frac{1 - λ}{{(1 - ζ_{0})}^{3}} \int_{R^{3}} F (T_{0}^{2} u_{1}^{\infty}) d x \\ + \frac{1 - ζ_{0}}{2} min_{1 - ζ_{0} ⩽ s ⩽ 1 + ζ_{0}} \int_{R^{3}} [V_{\infty} - V (s^{- 1} x)] {| u_{1}^{\infty} |}^{2} d x \\ > & c_{λ}, \forall λ \in (\bar{λ}, 1] . \end{array}$

Case ii). $t_{λ} \in (0, 1 - ζ_{0}) \cup (1 + ζ_{0}, T_{0})$ . From (V2), (4.2), (4.3), (4.8), (4.12), (4.15) and Lemma 4.4(iii)–(iv), we have $\begin{array}{rcl} m_{λ}^{\infty} & ⩾ & m_{1}^{\infty} = I_{1}^{\infty} (u_{1}^{\infty}) ⩾ I_{1}^{\infty} (t_{λ}^{2} {(u_{1}^{\infty})}_{t_{λ}}) + \frac{(1 - θ_{0}) V_{\infty} {(1 - t_{λ})}^{2} (2 + t_{λ})}{6} {‖ u_{1}^{\infty} ‖}_{2}^{2} \\ = & I_{λ} ({(u_{1}^{\infty})}_{t_{λ}}) - \frac{1 - λ}{t_{λ}^{3}} \int_{R^{3}} F (t_{λ}^{2} u_{1}^{\infty}) d x \\ + \frac{t_{λ}^{3}}{2} \int_{R^{3}} [V_{\infty} - V (t_{λ}^{- 1} x)] {| u_{1}^{\infty} |}^{2} d x + \frac{(1 - θ_{0}) V_{\infty} {(1 - t_{λ})}^{2} (2 + t_{λ})}{6} {‖ u_{1}^{\infty} ‖}_{2}^{2} \\ ⩾ & c_{λ} - \frac{1 - λ}{β_{0}^{3}} \int_{R^{N}} F (T_{0}^{2} u_{1}^{\infty}) d x + \frac{(1 - θ_{0}) V_{\infty} ζ_{0}^{2}}{3} {‖ u_{1}^{\infty} ‖}_{2}^{2} \\ > & c_{λ}, \forall λ \in (\bar{λ}, 1] . \end{array}$ In both cases, we obtain that $c_{λ} < m_{λ}^{\infty}$ for $λ \in (\bar{λ}, 1]$ . □
Lemma 4.6 (“splitting” of (PS) sequences, [30]).

Assume that (V1), (V2) and (F1)–(F3) hold. Let ${u_{n}}$ be a bounded ${(PS)}_{c_{λ}}$ sequence for $I_{λ}$ with $λ \in [1 / 2, 1]$ . Then there exist a subsequence of ${u_{n}}$ , still denoted by ${u_{n}}$ , an integer $l \in N \cup {0}$ , and $u_{λ} \in H^{1} (R^{3})$ such that

$u_{n} ⇀ u_{λ}$ in $H^{1} (R^{3})$ and $I_{λ}^{'} (u_{λ}) = 0$ ;

$w^{k} \neq 0$ and ${(I_{λ}^{\infty})}^{'} (w^{k}) = 0$ for $1 ⩽ k ⩽ l$ ;

$I_{λ} (u_{n}) \to I_{λ} (u_{λ}) + \sum_{i = 1}^{l} I_{λ}^{\infty} (w^{i})$ .

Here we agree that in the case

l = 0

the above holds without

w^{k}

Lemma 4.7.

Assume that (V1), (V2), (V4), (F1)–(F5) and (F6) hold. Then for almost every $λ \in [\bar{λ}, 1]$ , there exists a $u_{λ} \in H^{1} (R^{3}) ∖ {0}$ such that $\begin{matrix} (4.16) & I_{λ}^{'} (u_{λ}) = 0, I_{λ} (u_{λ}) = c_{λ} . \end{matrix}$

Proof.

Lemma 4.4 implies that $I_{λ} (u)$ satisfies the assumptions of Proposition 4.1 with $X = H^{1} (R^{3})$ and $Φ_{λ} = I_{λ}$ . So for almost every $λ \in [1 / 2, 1]$ , there exists a bounded sequence ${u_{n} (λ)} \subset H^{1} (R^{3})$ (for simplicity, we denote ${u_{n}}$ instead of ${u_{n} (λ)}$ such that $\begin{matrix} (4.17) & I_{λ} (u_{n}) \to c_{λ} > 0, ‖ I_{λ}^{'} (u_{n}) ‖ \to 0 . \end{matrix}$ By Lemma 4.6, there exist $l \in N \cup {0}$ and $u_{λ} \in H^{1} (R^{3})$ such that $I_{λ}^{'} (u_{λ}) = 0$ and $\begin{matrix} (4.18) & u_{n} ⇀ u_{λ} in H^{1} (R^{3}), I_{λ} (u_{n}) \to I_{λ} (u_{λ}) + \sum_{i = 1}^{l} I_{λ}^{\infty} (w^{i}), \end{matrix}$ where ${w^{i}}_{i = 1}^{l}$ are the critical points of $I_{λ}^{\infty}$ . Since $I_{λ}^{'} (u_{λ}) = 0$ , then $P_{λ} (u_{λ}) = 0$ . By Hardy inequality, one has $\begin{matrix} (4.19) & ‖ \nabla u ‖_{2}^{2} ⩾ \frac{1}{4} \int_{R^{3}} \frac{u^{2}}{| x |^{2}} d x, \forall u \in H^{1} (R^{3}) . \end{matrix}$ It follows from (V4), (F6), (4.2), (4.4) and (4.19) that $\begin{array}{rcl} I_{λ} (u_{λ}) & = & I_{λ} (u_{λ}) - \frac{1}{4 α - 5} [α ⟨ I_{λ}^{'} (u_{λ}), u_{λ} ⟩ - P_{λ} (u_{λ})] \\ = & \frac{α - 2}{4 α - 5} ‖ \nabla u_{λ} ‖_{2}^{2} + \frac{1}{2 (4 α - 5)} \int_{R^{3}} [2 (α - 1) V (x) + \nabla V (x) \cdot x] u_{λ}^{2} d x \\ + \frac{λ}{4 α - 5} \int_{R^{3}} [α f (u_{λ}) u_{λ} - 2 (2 α - 1) F (u_{λ})] d x \\ ⩾ & 0 . \end{array}$ If $l \neq 0$ , then $\begin{matrix} c_{λ} = lim_{n \to \infty} I_{λ} (u_{n}) = I_{λ} (u_{λ}) + \sum_{i = 1}^{l} I_{λ}^{\infty} (w^{i}) ⩾ m_{λ}^{\infty}, \forall λ \in (\bar{λ}, 1], \end{matrix}$ which is a contradiction by Lemma 4.5. Thus $l = 0$ and then (4.17) and Lemma 4.6 imply that $I_{λ} (u_{λ}) = c_{λ}$ . □

Lemma 4.8.

Assume that (V1), (V2), (V5), (F1)–(F5) and (F7) hold. Then for almost every $λ \in (\bar{λ}, 1]$ , there exists $u_{λ} \in H^{1} (R^{3}) ∖ {0}$ such that $\begin{matrix} (4.20) & I_{λ}^{'} (u_{λ}) = 0, I_{λ} (u_{λ}) = c_{λ} . \end{matrix}$

Proof.

In view of the proof of Lemma 4.7, for almost every $λ \in [0.5, 1]$ , there exists a bounded sequence ${u_{n} (λ)} \subset H^{1} (R^{3})$ (for simplicity, we denote the sequence by ${u_{n}}$ instead of ${u_{n} (λ)}$ ) such that $u_{n} ⇀ u_{λ}$ in $H^{1} (R^{3})$ and $I_{λ}^{'} (u_{λ}) = 0$ , and there exist $l \in N$ and $w^{1}, \dots, w^{l} \in H^{1} (R^{3}) ∖ {0}$ such that ${(I_{λ}^{\infty})}^{'} (w^{k}) = 0$ for $1 ⩽ k ⩽ l$ , moreover, (4.17) and (4.18) hold. By (V1), (V2) and (V5), there exists $α > 2$ such that $\begin{array}{rcl} \int_{R^{3}} [| \nabla u_{λ} |^{2} + V (x) u_{λ}^{2}] d x & ⩾ & ϱ_{0} \int_{R^{3}} u_{λ}^{2} d x \\ (4.21) & ⩾ & - \frac{1}{α - 2} \int_{R^{3}} [V (x) + \frac{1}{2} \nabla V (x) \cdot x] u_{λ}^{2} d x . \end{array}$ It follows from (F5), (F7), (4.2), (4.4) and (4.21) that $\begin{array}{rcl} I_{λ} (u_{λ}) & = & I_{λ} (u_{λ}) - \frac{1}{4 α - 5} [α ⟨ I_{λ}^{'} (u_{λ}), u_{λ} ⟩ - P_{λ} (u_{λ})] \\ = & \frac{α - 2}{4 α - 5} ‖ \nabla u_{λ} ‖_{2}^{2} + \frac{1}{2 (4 α - 5)} \int_{R^{3}} [2 (α - 1) V (x) + \nabla V (x) \cdot x] u_{λ}^{2} d x \\ + \frac{λ}{4 α - 5} \int_{R^{3}} [α f (u_{λ}) u_{λ} - 2 (2 α - 1) F (u_{λ})] d x \\ = & \frac{α - 2}{4 α - 5} \int_{R^{3}} [| \nabla u_{λ} |^{2} + V (x) u_{λ}^{2}] d x \\ + \frac{1}{4 α - 5} \int_{R^{3}} [V (x) + \frac{1}{2} \nabla V (x) \cdot x] u_{λ}^{2} d x \\ + \frac{λ}{4 α - 5} \int_{R^{3}} [α f (u_{λ}) u_{λ} - 2 (2 α - 1) F (u_{λ})] d x \\ ⩾ & \frac{λ}{4 α - 5} \int_{R^{3}} [α f (u_{λ}) u_{λ} - 2 (2 α - 1) F (u_{λ})] d x \\ ⩾ & \frac{[α (μ - 4) + 2] λ}{4 α - 5} \int_{R^{3}} F (u_{λ}) d x \\ (4.22) & ⩾ & 0 . \end{array}$ If $l \neq 0$ , then $\begin{matrix} c_{λ} = lim_{n \to \infty} I_{λ} (u_{n}) = I_{λ} (u_{λ}) + \sum_{i = 1}^{l} I_{λ}^{\infty} (w^{i}) ⩾ m_{λ}^{\infty}, \forall λ \in [\bar{λ}, 1], \end{matrix}$ which is a contradiction by Lemma 4.5. Thus $l = 0$ and then (4.17) and Lemma 4.6 imply that $I_{λ} (u_{λ}) = c_{λ}$ . □

Proof of Theorem 1.2.

In view of Lemma 4.2 and 4.7, there exist two sequences of ${λ_{n}} \subset (\bar{λ}, 1]$ and ${u_{λ_{n}}} \subset H^{1} (R^{3})$ , denoted by ${u_{n}}$ , such that $\begin{matrix} (4.23) & λ_{n} \to 1, I_{λ_{n}}^{'} (u_{n}) = 0, P_{λ_{n}} (u_{n}) = 0, I_{λ_{n}} (u_{n}) = c_{λ_{n}} . \end{matrix}$ From (V4), (F6), (4.2), (4.8), (4.19), (4.23) and Lemma 4.4(iii), one has $\begin{array}{rcl} c_{1 / 2} & ⩾ & c_{λ_{n}} = I_{λ_{n}} (u_{n}) - \frac{1}{4 α - 5} [α ⟨ I_{λ_{n}}^{'} (u_{n}), u_{n} ⟩ - P_{λ_{n}} (u_{n})] \\ = & \frac{α - 2}{4 α - 5} ‖ \nabla u_{n} ‖_{2}^{2} + \frac{1}{2 (4 α - 5)} \int_{R^{3}} [2 (α - 1) V (x) + \nabla V (x) \cdot x] u_{n}^{2} d x \\ + \frac{λ_{n}}{4 α - 5} \int_{R^{3}} [α f (u_{n}) u_{n} - 2 (2 α - 1) F (u_{n})] d x \\ (4.24) & ⩾ & \frac{(1 - θ_{0}) (α - 2)}{4 α - 5} ‖ \nabla u_{n} ‖_{2}^{2} . \end{array}$ This shows that ${‖ \nabla u_{n} ‖_{2}}$ is bounded. Next, we demonstrate that ${u_{n}}$ is bounded in $H^{1} (R^{3})$ . According to (V1), (V2), (F1), (F2), (4.2), (4.23), (4.24) and the Sobolev embedding theorem, we have $\begin{array}{rcl} ϱ_{0} ‖ u_{n} ‖_{2}^{2} & ⩽ & \int_{R^{3}} [| \nabla u_{n} |^{2} + V (x) u_{n}^{2}] d x \\ ⩽ & 2 c_{λ_{n}} + 2 λ_{n} \int_{R^{3}} F (u_{n}) d x \\ ⩽ & 2 c_{1 / 2} + \frac{ϱ_{0}}{4} ‖ u_{n} ‖_{2}^{2} + C_{2} ‖ u_{n} ‖_{6}^{6} \\ ⩽ & 2 c_{1 / 2} + \frac{ϱ_{0}}{4} ‖ u_{n} ‖_{2}^{2} + C_{2} S^{- 3} ‖ \nabla u_{n} ‖_{2}^{6} . \end{array}$ Hence, ${‖ u_{n} ‖_{2}}$ is bounded, and so ${u_{n}}$ is bounded in $H^{1} (R^{3})$ . In view of Lemma 4.4(v), we have ${lim}_{n \to \infty} c_{λ_{n}} = c_{*} ⩽ c_{1}$ . Hence, it follows from (1.3), (4.2), (4.9) and (4.23) that $\begin{matrix} (4.25) & I (u_{n}) \to c_{*} ⩾ κ_{0}, I^{'} (u_{n}) \to 0 . \end{matrix}$ This shows that ${u_{n}}$ satisfy (4.17) with $I_{λ} = I$ and $c_{λ} = c_{*}$ . In view of the proof of Lemma 4.7, we can show that there exists $\tilde{u} \in H^{1} (R^{3}) ∖ {0}$ such that $\begin{matrix} (4.26) & I^{'} (\tilde{u}) = 0, 0 < I (\tilde{u}) ⩽ c_{1} . \end{matrix}$ Let $\begin{matrix} K : = {u \in H^{1} (R^{3}) ∖ {0} : I^{'} (u) = 0}, \hat{m} : = inf_{u \in K} I (u) . \end{matrix}$ Then (4.26) shows that $K \neq \emptyset$ and $\hat{m} ⩽ c_{1}$ . For any $u \in K$ , Lemma 4.2 implies $\begin{array}{rcl} P (u) & : = & \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{1}{2} \int_{R^{3}} [3 V (x) + \nabla V (x) \cdot x] u^{2} d x \\ (4.27) & + \frac{5}{4} \int_{R^{3}} ϕ_{u} (x) u^{2} d x - 3 λ \int_{R^{3}} F (u) d x = 0 . \end{array}$ By (1.3), (V1), (V2), (F1), (F2) and the fact $⟨ I^{'} (u), u ⟩ = 0$ , it is easy to verify that $\begin{matrix} (4.28) & δ : = inf {‖ \nabla u ‖_{2} : u \in K} > 0 . \end{matrix}$ Then we deduce from (V4), (F6), (1.3), (4.27) and (4.28) that $\begin{array}{rcl} I (u) & = & I (u) - \frac{1}{4 α - 5} [α ⟨ I^{'} (u), u ⟩ - P (u)] \\ = & \frac{α - 2}{4 α - 5} ‖ \nabla u ‖_{2}^{2} + \frac{1}{2 (4 α - 5)} \int_{R^{3}} [2 (α - 1) V (x) + \nabla V (x) \cdot x] u^{2} d x \\ + \frac{1}{4 α - 5} \int_{R^{3}} [α f (u) u - 2 (2 α - 1) F (u)] d x \\ ⩾ & \frac{(1 - θ) (α - 2)}{4 α - 5} ‖ \nabla u ‖_{2}^{2} \\ (4.29) & ⩾ & \frac{(1 - θ) (α - 2)}{4 α - 5} δ^{2}, \forall u \in K . \end{array}$ Hence $\hat{m} > 0$ . Let ${u_{n}} \subset K$ such that $\begin{matrix} (4.30) & I^{'} (u_{n}) = 0, I (u_{n}) \to \hat{m} . \end{matrix}$ In view of Lemma 4.5, $\hat{m} ⩽ c_{1} < m_{1}^{\infty}$ . By a similar argument as in the proof of Lemma 4.7, we can prove that there exists $\bar{u} \in H^{1} (R^{N}) ∖ {0}$ such that $\begin{matrix} (4.31) & I^{'} (\bar{u}) = 0, I (\bar{u}) = \hat{m} > 0 . \end{matrix}$ This shows that $\bar{u}$ is a ground state solution of (1.1). □

Proof of Theorem 1.3.

The proof is similar to one of Theorem 1.2. So we omit it. □

Proof of Theorem 1.4.

In view of Lemmas 4.2 and 4.8, there exist two sequences ${λ_{n}} \subset (\bar{λ}, 1]$ and ${u_{λ_{n}}} \subset H^{1} (R^{3})$ , denoted by ${u_{n}}$ , such that $\begin{matrix} (4.32) & λ_{n} \to 1, I_{λ_{n}}^{'} (u_{n}) = 0, P_{λ_{n}}^{'} (u_{n}) = 0, I_{λ_{n}} (u_{n}) = c_{λ_{n}} . \end{matrix}$ By (V1), (V2) and (V5), there exists $α > 3$ such that $\begin{array}{rcl} \int_{R^{3}} [| \nabla u |^{2} + V (x) u^{2}] d x & ⩾ & ϱ_{0} \int_{R^{3}} u^{2} d x \\ ⩾ & - \frac{1}{α - 3} \int_{R^{3}} [2 V (x) + \frac{1}{2} \nabla V (x) \cdot x] u^{2} d x, \\ (4.33) & \forall u \in H^{1} (R^{3}) . \end{array}$ From (F7), (4.2), (4.4), (4.32) and (4.33), one has $\begin{array}{rcl} c_{1 / 2} & ⩾ & c_{λ_{n}} = I_{λ_{n}} (u_{n}) - \frac{1}{4 α - 5} [α ⟨ I_{λ_{n}}^{'} (u_{n}), u_{n} ⟩ - P_{λ_{n}} (u_{n})] \\ = & \frac{α - 2}{4 α - 5} ‖ \nabla u_{n} ‖_{2}^{2} + \frac{1}{2 (4 α - 5)} \int_{R^{3}} [2 (α - 1) V (x) + \nabla V (x) \cdot x u_{n}^{2}] d x \\ + \frac{λ_{n}}{4 α - 5} \int_{R^{3}} [α f (u_{n}) u_{n} - 2 (2 α - 1) F (u_{n})] d x \\ = & \frac{1}{4 α - 5} ‖ \nabla u_{n} ‖_{2}^{2} + \frac{α - 3}{4 α - 5} \int_{R^{3}} [| \nabla u_{n} |^{2} + V (x) u_{n}^{2}] d x \\ + \frac{1}{4 α - 5} \int_{R^{3}} [2 V (x) + \frac{1}{2} \nabla V (x) \cdot x] u_{n}^{2} d x \\ + \frac{λ_{n}}{4 α - 5} \int_{R^{3}} [α f (u_{n}) u_{n} - 2 (2 α - 1) F (u_{n})] d x \\ ⩾ & \frac{1}{4 α - 5} ‖ \nabla u_{n} ‖_{2}^{2} + \frac{[α (μ - 4) + 2] λ_{n}}{4 α - 5} \int_{R^{3}} F (u_{n}) d x \\ (4.34) & ⩾ & \frac{1}{4 α - 5} ‖ \nabla u_{n} ‖_{2}^{2} . \end{array}$ This shows that ${‖ \nabla u_{n} ‖_{2}}$ is bounded. Similarly to the proof of Theorem 1.1, we can show that there exists $\tilde{u} \in H^{1} (R^{3}) ∖ {0}$ such that $\begin{matrix} (4.35) & I^{'} (\tilde{u}) = 0, 0 < I (\tilde{u}) ⩽ c_{1} . \end{matrix}$ Let $\begin{matrix} K : = {u \in H^{1} (R^{3}) ∖ {0} : I^{'} (u) = 0}, \hat{m} : = inf_{u \in K} I (u) . \end{matrix}$ Then (4.35) shows that $K \neq \emptyset$ and $\hat{m} ⩽ c_{1}$ . By (1.3), (V1), (V2), (F1), (F2) and the fact $⟨ I^{'} (u), u ⟩ = 0$ , it is easy to verify that $\begin{matrix} (4.36) & δ : = inf {‖ \nabla u ‖_{2} : u \in K} > 0 . \end{matrix}$ From (F7), (1.3), (4.27), (4.33) and (4.36), we have $\begin{array}{rcl} I (u) & = & I (u) - \frac{1}{4 α - 5} [α ⟨ I^{'} (u), u ⟩ - P (u)] \\ = & \frac{α - 2}{4 α - 5} ‖ \nabla u ‖_{2}^{2} + \frac{1}{2 (4 α - 5)} \int_{R^{3}} [2 (α - 1) V (x) + \nabla V (x) \cdot x] u^{2} d x \\ + \frac{1}{4 α - 5} \int_{R^{3}} [α f (u) u - 2 (2 α - 1) F (u)] d x \\ = & \frac{1}{4 α - 5} ‖ \nabla u ‖_{2}^{2} + \frac{α - 3}{4 α - 5} \int_{R^{3}} [| \nabla u |^{2} + V (x) u^{2}] d x \\ + \frac{1}{4 α - 5} \int_{R^{3}} [2 V (x) + \frac{1}{2} \nabla V (x) \cdot x] u^{2} d x \\ + \frac{1}{4 α - 5} \int_{R^{3}} [α f (u) u - 2 (2 α - 1) F (u)] d x \\ ⩾ & \frac{1}{4 α - 5} ‖ \nabla u ‖_{2}^{2} + \frac{[α (μ - 4) + 2]}{4 α - 5} \int_{R^{3}} F (u) d x \\ (4.37) & ⩾ & \frac{1}{4 α - 5} ‖ \nabla u ‖_{2}^{2} ⩾ \frac{δ}{4 α - 5}, \forall u \in K . \end{array}$ Hence $\hat{m} > 0$ . The rest of the proof is the same as in the one of Theorem 1.1. □

Footnotes

Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (No: 11971485), Research Foundation of Education Bureau of Hunan Province, China (No. 18A452, No. 19B450) and the Huaihua University Double First-Class initiative Applied Characteristic Discipline of Control Science and Engineering.

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Ground state solutions of Schrödinger–Poisson systems with asymptotically constant potential

Abstract

Keywords

1. Introduction

Lemma 2.1. Assume that (F1), (F2) and (F5) hold. Then I ∞ ( u ) ⩾ I ∞ ( t 2 u t ) + 1 − t 3 3 J ∞ ( u ) (2.8) + ( 1 − θ 0 ) V ∞ ( 1 − t ) 2 ( 2 + t ) 6 ‖ u ‖ 2 2 , ∀ u ∈ H 1 ( R 3 ) , t ⩾ 0 . Lemma 2.2 ([30]).

Lemma 2.3 ([30]).

Lemma 2.4. Assume that V satisfies (V1), (V2). Then there exist constants γ 1 , γ 2 > 0 such that γ 1 ‖ u ‖ 2 ⩽ ∫ R 3 [ | ∇ u | 2 + V ( x ) u 2 ] d x ⩽ γ 2 ‖ u ‖ 2 , ∀ u ∈ H 1 ( R 3 ) . 3. Ground state solutions for (1.5)

Proposition 3.1 ([21]).

Lemma 3.2 ([29]).

Proposition 4.1 ([20]).

Lemma 4.2 ([30]).

Footnotes

Acknowledgements

References