Non-trivial solutions for the fractional Schrödinger

Abstract

In this paper, we study a fractional Schrödinger–Poisson system with p-Laplacian. By using some scaling transformation and cut-off technique, the boundedness of the Palais–Smale sequences at the mountain pass level is gotten. As a result, the existence of non-trivial solutions for the system is obtained.

Keywords

Fractional Schrödinger–Poisson system mountain pass lemma

1. Introduction and main results

In this paper, our purpose is to obtain the existence of non-trivial solutions in the following Schrödinger–Poisson system with fractional p-Laplacian: $\begin{matrix} (1.1) & \{\begin{array}{l} {(- Δ)}_{p}^{s} u + | u |^{p - 2} u + λ ϕ u = | u |^{q - 2} u, in R^{3}, \\ {(- Δ)}^{t} ϕ = u^{2}, in R^{3}, \end{array} \end{matrix}$ where $λ > 0$ , $s, t \in (0, 1)$ , $\frac{12}{3 + 2 t + 4 s} < p < \frac{12}{3 + 2 t}$ , $p < q < p_{s}^{*} : = \frac{3 p}{3 - p s}$ , $p s < 3$ and ${(- Δ)}_{p}^{s}$ is the fractional p-Laplacian given by $\begin{matrix} {(- Δ)}_{p}^{s} u (x) = C_{s} P . V . \int_{R^{3}} \frac{| u (x) - u (y) |^{p - 2} (u (x) - u (y))}{| x - y |^{3 + p s}} d y, \end{matrix}$ where $C_{s}$ is a normalized constant and $P . V$ . stands for the Cauchy principal value. The main results of this paper are listed in following.

Theorem 1.1.
Suppose that $q \in (\frac{(2 t + 4 s) p}{2 t + p s}, p_{s}^{})$ . Then for each fixed* $λ > 0$ , system ( 1.1 ) admits at least one non-trivial solution.
Theorem 1.2.
Suppose that $q \in (p, \frac{(2 t + 4 s) p}{2 t + p s}]$ . Then there exists $λ^{} > 0$ such that for each* $λ \in (0, λ^{})$ , system (* 1.1 ) admits at least one non-trivial solution.

Recently many researchers focus on the investigation of the problems about non-local equations (or system) in mathematical analysis field. These problems can more accurately describe some physical or chemical phenomena. About the background of this topic one can refer [5–7,13,17,21,22] and the references therein. In particular, when $λ = 0$ the first equation of the Schrödinger system (1.1) involving a non-local operator is a kind of fractional Schrödinger equation with p-Laplacian, for instance, Torres et al [27] and Zhao et al [34] had established the existence of nontrivial solution by applying mountain pass theorem with Cerami condition and by proving the compactness results using the fractional version of concentration compactness principle respectively. If $p = 2$ , this equation reduces to a fractional Schrödinger equation with Laplacian, such as, Zuo et al [35] develops a variational approach based on the scaling function method to establish the normalized solutions, but in [2], the authors considered the existence of entropy solutions to local nonlinear Schrödinger-type equations in the context of Sobolev spaces with variable exponents. The fractional Schrödinger equation which was firstly introduced by Laskin [17] has been studied extensively in the past decades. For instance, one can refer the recent papers [1,20] and the references therein.

If $p = 2$ , $s = 1$ , $t = 1$ , system (1.1) reduces to the Schrödinger–Poisson system $\begin{matrix} (1.2) & \{\begin{array}{l} - Δ u + u + λ ϕ u = | u |^{q - 2} u, in R^{3}, \\ - Δ ϕ = u^{2}, in R^{3} . \end{array} \end{matrix}$ Since the original work was established by Benci and Fortunato in [3,4], the systems as (1.2) have attracted increasing attentions in recent years, see [8,9,14,18,28,29] and the references therein, for example. Furthermore, as far as the authors know, Du et al in [11] first considered the existence of non-trivial solution for the following Schrödinger–Poisson system with p-Laplacian $\begin{matrix} (1.3) & \{\begin{array}{l} - Δ_{p} u + | u |^{p - 2} u + λ ϕ u = | u |^{q - 2} u, in R^{3}, \\ - Δ ϕ = u^{2}, in R^{3}, \end{array} \end{matrix}$ where $λ > 0$ , $\frac{4}{3} < p < \frac{12}{5}$ , $p < q < p^{} : = \frac{3 p}{3 - p}$ and $Δ_{p} u = div (| \nabla u |^{p - 2} \nabla u)$ is the p-Laplacian.

In the past decades, fractional Schrödinger–Poisson systems have received considerable attention. For example, Feng in [12] studied the following Schrödinger–Poisson system $\begin{matrix} (1.4) & \{\begin{array}{l} - Δ u + u + ϕ u = f (x, u), in R^{3}, \\ {(- Δ)}^{\frac{α}{2}} ϕ = u^{2}, {lim}_{| x | \to \infty} ϕ (x) = 0, in R^{3}, \end{array} \end{matrix}$ where $α \in (1, 2]$ . It was proved that system (1.4) possesses at least one non-trivial weak solution under some conditions on f. Li in [19] considered the following system $\begin{matrix} (1.5) & \{\begin{array}{l} {(- Δ)}^{s} u + u + ϕ u = f (x, u), in R^{3}, \\ {(- Δ)}^{t} ϕ = u^{2}, in R^{3}, \end{array} \end{matrix}$ where $s, t \in (0, 1]$ , $2 t + 4 s > 3$ . Under some suitable conditions on f, it was proved that the system (1.5) admits at least one non-trivial weak solution by using the perturbation method and the mountain pass lemma. Teng in [26] considered the existence of ground state solutions for the following system $\begin{matrix} (1.6) & \{\begin{array}{l} {(- Δ)}^{s} u + V (x) u + ϕ u = μ | u |^{q - 2} u + | u |^{2_{s}^{} - 2} u, in R^{3}, \\ {(- Δ)}^{t} ϕ = u^{2}, in R^{3}, \end{array} \end{matrix}$ where $μ ⩾ 0$ , $1 < q < 2_{s}^{} - 1 = \frac{3 + 2 s}{3 - 2 s}, s$ , $t \in (0, 1)$ , $2 t + 2 s > 3$ . Zhang et al* in [33] established the existence of positive solutions for the following system $\begin{matrix} (1.7) & \{\begin{array}{l} {(- Δ)}^{s} u + λ ϕ u = g (x, u), in R^{3}, \\ {(- Δ)}^{t} ϕ = λ u^{2}, in R^{3}, \end{array} \end{matrix}$ where $s, t \in (0, 1]$ and g is a nonlinearity of Berestycki–Lions type. Yu et al in [32] studied the following system with critical growth $\begin{matrix} (1.8) & \{\begin{array}{l} {(- Δ)}^{s} u + u + k (x) ϕ u = | u |^{2^{} - 2} u + h (x) | u |^{p - 2} u, in R^{3}, \\ {(- Δ)}^{t} ϕ = k (x) u^{2}, in R^{3}, \end{array} \end{matrix}$ where $s \in (\frac{3}{4}, 1)$ , $t \in (0, 1)$ . It was shown that there exist a positive and a sign-changing solutions to system (1.8).

This paper is organized in three sections. In Section 2, besides some important lemmas, the variational setting for (1.1) will be established. In Section 3, by using the famous mountain pass lemma in [24], we give a proof of Theorem 1.1 and Theorem 1.2. The key point of the proofs focuses on the boundedness of Palais–Smale (PS for short) sequence at the mountain pass level. We should mention that when $4 ⩽ q < p_{s}^{}$ , it is easy to check that any (PS) sequence at the mountain pass level is bounded. For the case $p < q < 4$ , in Section 3 of this paper, we divide it into two cases with $\frac{(2 t + 4 s) p}{2 t + p s} < q < p_{s}^{*}$ and $p < q ⩽ \frac{(2 t + 4 s) p}{2 t + p s}$ by applying the scaling transformation as in [15] and the cut-off method as in [16] to get a bounded subsequence of (PS) sequence at the mountain pass level, respectively.
2. Variational setting

First of all, we introduce some useful notations associated to the fractional Sobolev space $W^{s, p} (R^{3})$ as follows. For $p \in [1, + \infty)$ , we denote by $L^{p} (R^{3})$ the usual Lebesgue space with the norm $‖ u ‖_{p} = {(\int_{R^{3}} | u |^{p} d x)}^{\frac{1}{p}}$ . And the Gagliardo seminorm is given by $\begin{matrix} {[u]}_{s, p} : = {(\int_{R^{3}} \int_{R^{3}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{3 + p s}} d x d y)}^{\frac{1}{p}} . \end{matrix}$ The fractional Sobolev space $W^{s, p} (R^{3})$ is defined by $\begin{matrix} W : = W^{s, p} (R^{3}) = {u \in L^{p} (R^{3}) : {[u]}_{s, p} < + \infty}, \end{matrix}$ and is equipped with the norm $\begin{matrix} ‖ u ‖ : = {({[u]}_{s, p}^{p} + ‖ u ‖_{p}^{p})}^{\frac{1}{p}} . \end{matrix}$

Next, we recall the following fractional Sobolev inequality and Sobolev embedding theorem for our subsequent uses. In the sequel of this paper, we denote by C the positive constant, which may change from line to line.

Lemma 2.1 ([23]).

Let $s \in (0, 1)$ and $p \in [1, + \infty)$ be such that $p s < 3$ . Then there exists a constant $C > 0$ , dependent of p, s, such that $\begin{matrix} ‖ u ‖_{p_{s}^{*}}^{p} ⩽ C {[u]}_{s, p}^{p} . \end{matrix}$ Therefore, the fractional Sobolev space W is continuously embedded in $L^{q} (R^{3})$ for any $q \in [p, p_{s}^{*}]$ . Besides the embedding $W ↪ L^{q} (R^{3})$ is locally compact for $q \in [p, p_{s}^{*})$ .

Let $t \in (0, 1)$ and $2_{t}^{*} : = \frac{6}{3 - 2 t}$ , the fractional Sobolev space $D^{t, 2} (R^{3})$ is given as follows $\begin{matrix} D^{t, 2} (R^{3}) = {u \in L^{2_{t}^{*}} (R^{3}) : {[u]}_{t, 2} < + \infty}, \end{matrix}$ which is the completion of $C_{0}^{\infty} (R^{3})$ with respect to the norm $\begin{matrix} {[u]}_{t, 2} : = {(\int_{R^{3}} \int_{R^{3}} \frac{| u (x) - u (y) |^{2}}{| x - y |^{3 + 2 t}} d x d y)}^{\frac{1}{2}} . \end{matrix}$

Let $s \in (0, 1)$ and $1 < p < + \infty$ . Then W is a reflexive Banach space. For more details about fractional Sobolev spaces and the fractional Laplacian, one can refer to [23].

The dual space of $(W, ‖ \cdot ‖)$ is denoted by $(W^{*}, ‖ \cdot ‖_{*})$ . We rephrase variationally the fractional p-Laplacian as the nonlinear operator $A : W \to W^{*}$ defined for any $u, v \in W$ by $\begin{matrix} ⟨ A (u), v ⟩ = \int_{R^{3}} \int_{R^{3}} \frac{| u (x) - u (y) |^{p - 2} (u (x) - u (y)) (v (x) - v (y))}{| x - y |^{3 + 2 s}} d x d y + \int_{R^{3}} | u |^{p - 2} u v d x . \end{matrix}$

By Lemma 2.1, we get $W ↪ L^{\frac{12}{3 + 2 t}} (R^{3})$ for $\frac{12}{3 + 2 t + 4 s} < p < \frac{12}{3 + 2 t}$ . The linear operator $T_{u} : D^{t, 2} (R^{3}) \to R$ is given by $\begin{matrix} T_{u} (v) = \int_{R^{3}} u^{2} v d x, \forall u \in W . \end{matrix}$ It follows from Hölder’s inequality and Lemma 2.1 that $\begin{matrix} (2.1) & | T_{u} (v) | ⩽ ‖ u ‖_{\frac{12}{3 + 2 t}}^{2} ‖ v ‖_{2_{t}^{*}} ⩽ C ‖ u ‖^{2} {[v]}_{t, 2} . \end{matrix}$ Suppose $\begin{matrix} η (u, v) = \int_{R^{3}} \int_{R^{3}} \frac{(u (x) - u (y)) (v (x) - v (y))}{| x - y |^{3 + 2 t}} d x d y, \forall u, v \in D^{t, 2} (R^{3}) . \end{matrix}$ Then one can easily check that $η (u, v)$ is bilinear, bounded and coercive. By the Lax–Milgram theorem, for each $u \in W$ one can see that there exists a unique $ϕ_{u}^{t} \in D^{t, 2} (R^{3})$ such that $T_{u} (v) = η (ϕ_{u}^{t}, v)$ for all $v \in D^{t, 2} (R^{3})$ , i.e. $\begin{matrix} (2.2) & \int_{R^{3}} \int_{R^{3}} \frac{(ϕ_{u}^{t} (x) - ϕ_{u}^{t} (y)) (v (x) - v (y))}{| x - y |^{3 + 2 t}} d x d y = \int_{R^{3}} u^{2} v d x . \end{matrix}$ Hence ${(- Δ)}^{t} ϕ_{u}^{t} = u^{2}$ in weak sense. Then it is easy to verify that $Φ : W \to D^{t, 2} (R^{3})$ given as $\begin{matrix} (2.3) & Φ (u) = ϕ_{u}^{t} \end{matrix}$ is well defined. From (2.1) and (2.2), we get $\begin{matrix} (2.4) & {[ϕ_{u}^{t}]}_{t, 2} = ‖ T_{u} ‖ ⩽ C ‖ u ‖^{2} . \end{matrix}$ Since $s, t \in (0, 1)$ , $\frac{12}{3 + 2 t + 4 s} < p < \frac{12}{3 + 2 t}$ and $p s < 3$ , then $\frac{12}{3 + 2 t} \in (p, p_{s}^{*})$ . It follows from (2.1), (2.2) and Lemma 2.1 that $\begin{aligned} {[ϕ_{u}^{t}]}_{t, 2}^{2} = & \int_{R^{3}} \int_{R^{3}} \frac{| ϕ_{u}^{t} (x) - ϕ_{u}^{t} (y) |^{2}}{| x - y |^{3 + 2 t}} d x d y \\ (2.5) & = & \int_{R^{3}} ϕ_{u}^{t} u^{2} d x ⩽ ‖ u ‖_{\frac{12}{3 + 2 t}}^{2} {‖ ϕ_{u}^{t} ‖}_{2_{t}^{*}} ⩽ C ‖ u ‖_{\frac{12}{3 + 2 t}}^{2} {[ϕ_{u}^{t}]}_{t, 2}, \end{aligned}$ which implies $\begin{aligned} (2.6) & {[ϕ_{u}^{t}]}_{t, 2} ⩽ C ‖ u ‖_{\frac{12}{3 + 2 t}}^{2} . \end{aligned}$ From [25], $ϕ_{u}^{t}$ is the Riesz potential given as $\begin{aligned} (2.7) & ϕ_{u}^{t} (x) = c_{t} \int_{R^{3}} \frac{u^{2} (y)}{| x - y |^{3 - 2 t}} d y, \forall x \in R^{3}, \end{aligned}$ where $\begin{matrix} c_{t} = \frac{Γ (\frac{3 - 2 t}{2})}{π^{\frac{3}{2}} 2^{2 t} Γ (t)} . \end{matrix}$ Substituting $ϕ_{u}^{t}$ in (1.1), we get the following fractional Schrödinger equation with p-Laplacian $\begin{matrix} (2.8) & {(- Δ)}_{p}^{s} u + | u |^{p - 2} u + λ ϕ_{u}^{t} u = | u |^{q - 2} u, in R^{3} . \end{matrix}$

Lemma 2.2.
Let ${u_{k}}$ be bounded in W and satisfy $\begin{matrix} (2.9) & sup_{y \in R^{3}} \int_{B_{σ} (y)} | u_{k} |^{p} d x \to 0 as k \to + \infty, \end{matrix}$ where $σ > 0$ and $B_{σ} (y) : = {x \in R^{3} : | x - y | < σ}$ . Then $u_{k} \to 0$ in $L^{r} (R^{3})$ for $r \in (p, p_{s}^{})$ .
Proof.
Let $τ \in (p, p_{s}^{})$ , it follows form interpolation inequality that $\begin{matrix} ‖ u_{k} ‖_{L^{τ} (B_{σ} (y))} ⩽ ‖ u_{k} ‖_{L^{p} (B_{σ} (y))}^{1 - θ} ‖ u_{k} ‖_{L^{p_{s}^{}} (B_{σ} (y))}^{θ}, \end{matrix}$ where $\begin{matrix} \frac{1}{τ} = \frac{1 - θ}{p} + \frac{θ}{p_{s}^{}} . \end{matrix}$ Now by choosing a covering of $R^{3}$ with balls of radius σ such that each point of $R^{3}$ is contained in at most 4 balls. Then we get $\begin{matrix} ‖ u_{k} ‖_{τ} ⩽ C sup_{y \in R^{3}} {(\int_{B_{σ} (y)} | u_{k} |^{p} d x)}^{\frac{1 - θ}{p}} ‖ u_{k} ‖_{p_{s}^{}}^{θ} ⩽ C sup_{y \in R^{3}} {(\int_{B_{σ} (y)} | u_{k} |^{p} d x)}^{\frac{1 - θ}{p}} ‖ u_{k} ‖^{θ} . \end{matrix}$ In the last inequality we have used Lemma 2.1. Hence we have $u_{k} \to 0$ in $L^{τ} (R^{3})$ for $τ \in (p, p_{s}^{})$ . □

The properties of $ϕ_{u}^{t}$ are as follows.
Proposition 2.3.
Let $u \in W$ , then we have
${[ϕ_{u}^{t}]}_{t, 2} ⩽ C ‖ u ‖^{2}$ with C independent of u;

for any $l > 0$ , $ϕ_{l u}^{t} = l^{2} ϕ_{u}^{t}$ and $ϕ_{u_{l}}^{t} (x) = l^{2 k - 2 t} ϕ_{u}^{t} (l x)$ with $u_{l} (x) = l^{k} u (l x)$ , $\forall k \in R$ ;

if $u_{n} ⇀ u$ in W, then $ϕ_{u_{n}}^{t} ⇀ ϕ_{u}^{t}$ in $D^{t, 2} (R^{3})$ and $\begin{matrix} (2.10) & \int_{R^{3}} ϕ_{u_{n}}^{t} u_{n} v d x \to \int_{R^{3}} ϕ_{u}^{t} u v d x for all v \in W . \end{matrix}$

Proof.
By using (2.4) and (2.7), one can get the proof of (i) and (ii), respectively. For (iii), let ${u_{n}} \subset W$ be such that $u_{n} ⇀ u$ in W. For any $ν \in C_{0}^{\infty} (R^{3})$ , we get $\begin{matrix} η (ϕ_{u_{n}}^{t} - ϕ_{u}^{t}, ν) = \int_{R^{3}} (u_{n}^{2} - u^{2}) ν d x . \end{matrix}$ It follows from the generalization of Hölder’s inequality that $\begin{aligned} \int_{R^{3}} (u_{n}^{2} - u^{2}) ν d x & ⩽ {(\int_{Ω} | u_{n} - u |^{\frac{12}{3 + 2 t}} d x)}^{\frac{3 + 2 t}{12}} {(\int_{Ω} | u_{n} + u |^{\frac{12}{3 + 2 t}} d x)}^{\frac{3 + 2 t}{12}} {(\int_{Ω} | ν |^{\frac{6}{3 - 2 t}} d x)}^{\frac{3 - 2 t}{6}} \\ ⩽ C {(\int_{Ω} | u_{n} - u |^{\frac{12}{3 + 2 t}} d x)}^{\frac{3 + 2 t}{12}} {(\int_{R^{3}} | u_{n} + u |^{\frac{12}{3 + 2 t}} d x)}^{\frac{3 + 2 t}{12}} ‖ ν ‖_{\infty}, \end{aligned}$ where Ω is the support of ν. Since $\frac{12}{3 + 2 t} \in (p, p_{s}^{})$ and the density of $C_{0}^{\infty} (R^{3})$ in $D^{t, 2} (R^{3})$ , hence by the above inequality and Lemma 2.1, we obtain $ϕ_{u_{n}}^{t} ⇀ ϕ_{u}^{t}$ in $D^{t, 2} (R^{3})$ . This means that $\begin{matrix} \int_{R^{3}} ϕ_{u_{n}}^{t} u ν d x \to \int_{R^{3}} ϕ_{u}^{t} u ν d x as n \to + \infty \end{matrix}$ for all $ν \in C_{0}^{\infty} (R^{3})$ . By the generalization of Hölder’s inequality and Lemma 2.1, we get that $\begin{matrix} | \int_{R^{3}} ϕ_{u_{n}}^{t} (u_{n} - u) ν d x | ⩽ {‖ ϕ_{u_{n}}^{t} ‖}_{2_{t}^{}} ‖ u_{n} - u ‖_{L^{\frac{12}{3 + 2 t}} (Ω)} ‖ ν ‖_{L^{\frac{12}{3 + 2 t}} (Ω)} \to 0 as n \to + \infty . \end{matrix}$ Hence, we obtain $\begin{matrix} | \int_{R^{3}} ϕ_{u_{n}}^{t} u_{n} ν d x - \int_{R^{3}} ϕ_{u}^{t} u ν d x | ⩽ | \int_{R^{3}} (ϕ_{u_{n}}^{t} - ϕ_{u}^{t}) u ν d x | + | \int_{R^{3}} ϕ_{u_{n}}^{t} (u_{n} - u) ν d x | \to 0 \end{matrix}$ as $n \to + \infty$ , for any $ν \in C_{0}^{\infty} (R^{3})$ . Since the space $C_{0}^{\infty} (R^{3})$ is dense in W, hence (2.10) holds. We finish the proof. □

Now motivated by the ideas of [3,4], the variational setting for (1.1) will be established. We denote $F : W \times D^{t, 2} (R^{3}) \to R$ as $\begin{matrix} F (u, ϕ) = \frac{1}{p} ‖ u ‖^{p} + \frac{λ}{2} \int_{R^{3}} ϕ u^{2} d x - \frac{λ}{4} \int_{R^{3}} \int_{R^{3}} \frac{| ϕ (x) - ϕ (y) |^{2}}{| x - y |^{3 + 2 t}} d x d y - \frac{1}{q} ‖ u ‖_{q}^{q} . \end{matrix}$ Then we get $\begin{aligned} ⟨ \partial_{u} F (u, ϕ), v ⟩ = ⟨ A (u), v ⟩ + λ \int_{R^{3}} ϕ u v d x - \int_{R^{3}} | u |^{q - 2} u v d x and \\ ⟨ \partial_{ϕ} F (u, ϕ), ξ ⟩ = \frac{λ}{2} \int_{R^{3}} ξ u^{2} d x - \frac{λ}{2} \int_{R^{3}} \int_{R^{3}} \frac{(ϕ (x) - ϕ (y)) (ξ (x) - ξ (y))}{| x - y |^{3 + 2 t}} d x d y \end{aligned}$ for $v \in W$ and $ξ \in D^{t, 2} (R^{3})$ . Apparently, $F \in C^{1} (W \times D^{t, 2} (R^{3}), R)$ and its critical points are corresponding to the weak solutions of system (1.1). From (2.3), we know that the graph of Φ $\begin{matrix} G_{Φ} = {(u, ϕ) \in W \times D^{t, 2} (R^{3}) : \partial_{ϕ} F (u, ϕ) = 0} \end{matrix}$ is well defined. In addition, similar to Proposition 2.1 in [10], we see that Φ is $C^{1}$ on W from the implicit function theorem. Now let us define a functional $I$ as $\begin{matrix} I (u) = F (u, ϕ_{u}^{t}) . \end{matrix}$ Then we get $\begin{matrix} I (u) = \frac{1}{p} ‖ u ‖^{p} + \frac{λ}{4} \int_{R^{3}} ϕ_{u}^{t} u^{2} d x - \frac{1}{q} ‖ u ‖_{q}^{q} . \end{matrix}$ Therefore $I \in C^{1} (W, R)$ and for all $u \in W$ $\begin{aligned} (2.11) & ⟨ I^{'} (u), v ⟩ = ⟨ A (u), v ⟩ + λ \int_{R^{3}} ϕ_{u}^{t} u v d x - \int_{R^{3}} | u |^{q - 2} u v d x, \forall v \in W . \end{aligned}$

From the above discussion, we give the following definition. Definition 2.4.

We say u is a weak solution of (2.8) if $⟨ I^{'} (u), v ⟩ = 0$ for all $v \in W$ .

We say $(u, ϕ_{u}^{t}) \in W \times D^{t, 2} (R^{3})$ is a weak solution of system (1.1) if u is a weak solution of (2.8).

3. Proofs of main results

In this section, with the help of the mountain pass lemma [24], we verify Theorem 1.1 and Theorem 1.2.

It is well known that if ${u_{n}} \subset W$ is bounded, then there exists a renamed subsequence of ${u_{n}}$ converging to some u weakly in W, i.e. $u_{n} ⇀ u$ in W.

Lemma 3.1.

Let ${u_{n}} \subset W$ be bounded, then there exists a subsequence ${u_{n_{k}}}$ such that $u_{n_{k}} ⇀ u \in W$ . Moreover, $\begin{aligned} (3.1) & ⟨ A (u_{n_{k}}) - A (u), v ⟩ \to 0, \forall v \in W . \end{aligned}$

Proof.

It is only need to verify (3.1). Denote $⟨ A (u), v ⟩ = ⟨ A_{1} (u), v ⟩ + ⟨ A_{2} (u), v ⟩$ where $\begin{matrix} ⟨ A_{1} (u), v ⟩ = \int_{R^{3}} \int_{R^{3}} \frac{| u (x) - u (y) |^{p - 2} (u (x) - u (y)) (v (x) - v (y))}{| x - y |^{3 + 2 s}} d x d y \end{matrix}$ and $\begin{matrix} ⟨ A_{2} (u), v ⟩ = \int_{R^{3}} | u |^{p - 2} u v d x . \end{matrix}$ Since ${u_{n}}$ is bounded in W, then there exist a subsequence ${u_{n_{k}}}$ and u in W such that $\begin{matrix} (3.2) & \{\begin{array}{ll} u_{n_{k}} ⇀ u & in W, \\ u_{n_{k}} \to u & in L_{loc}^{q} (R^{3}), p ⩽ q < p_{s}^{*}, \\ u_{n_{k}} \to u & a.e. x \in R^{3}, \end{array} \end{matrix}$ as $k \to + \infty$ . Denoting by $p^{'} = p / (p - 1)$ the Hölder conjugate of p, then $| u_{n_{k}} (x) - u_{n_{k}} (y) |^{p - 2} (u_{n_{k}} (x) - u_{n_{k}} (y)) / | x - y |^{(3 + p s) / p^{'}}$ is bounded in $L^{p^{'}} (R^{6})$ and converges to $| u (x) - u (y) |^{p - 2} (u (x) - u (y)) / | x - y |^{(3 + p s) / p^{'}}$ a.e. in $R^{6}$ , and $(v (x) - v (y)) / | x - y |^{(3 + p s) / p} \in L^{p} (R^{6})$ , hence we obtain $\begin{aligned} (3.3) & ⟨ A_{1} (u_{n_{k}}), v ⟩ \to ⟨ A_{1} (u), v ⟩, \forall v \in W . \end{aligned}$ Moreover, it follows form (3.2) and Proposition 5.4.7 of [31] that $\begin{aligned} (3.4) & ⟨ A_{2} (u_{n_{k}}), v ⟩ \to ⟨ A_{2} (u), v ⟩, \forall v \in W . \end{aligned}$ By (3.3) and (3.4), we obtain (3.1). We finish the proof. □

Combining Lemma 2.2 and Lemma 3.1, we can get the following result which plays an important role in looking for the non-trivial solution of system (1.1).

Lemma 3.2.

Let ${u_{n}} \subset W$ be bounded such that $\begin{aligned} (3.5) & I (u_{n}) \to c > 0 and I^{'} (u_{n}) \to 0, as n \to + \infty . \end{aligned}$ Then there is $\bar{u} \in W ∖ {0}$ satisfying $I^{'} (\bar{u}) = 0$ .

Proof.

Based on ${u_{n}}$ is bounded in W, we see that as $n \to + \infty$ (in the sense of subsequence) $\begin{aligned} (3.6) & sup_{y \in R^{3}} \int_{B_{σ} (y)} | u_{n} |^{p} d x \to μ \end{aligned}$ for some $μ ⩾ 0$ . When $μ = 0$ , then by Lemma 2.2, we get $u_{n} \to 0$ in $L^{r} (R^{3})$ for any $r \in (p, p_{s}^{*})$ . Hence we get $\begin{aligned} (3.7) & \int_{R^{3}} | u_{n} |^{q} d x \to 0 as n \to + \infty, \end{aligned}$ where $p < q < p_{s}^{*}$ . Since $\frac{12}{3 + 2 t} \in (p, p_{s}^{*})$ , by the Hölder inequality and (i) of Proposition 2.3, we deduce that $\begin{aligned} (3.8) & \int_{R^{3}} ϕ_{u_{n}}^{t} u_{n}^{2} d x ⩽ C {(\int_{R^{3}} | u_{n} |^{\frac{12}{3 + 2 t}} d x)}^{\frac{3 + 2 t}{6}} \to 0 as n \to + \infty . \end{aligned}$ Then combining (3.7) and (3.8), we infer that $\begin{matrix} I (u_{n}) - \frac{1}{p} ⟨ I^{'} (u_{n}), u_{n} ⟩ = (\frac{λ}{4} - \frac{λ}{p}) \int_{R^{3}} ϕ_{u_{n}}^{t} u_{n}^{2} d x + (\frac{1}{p} - \frac{1}{q}) \int_{R^{3}} | u_{n} |^{q} d x \to 0, \end{matrix}$ which contradicts (3.5). Thus $μ > 0$ . From (3.6), there exists ${y_{n}} \subset R^{3}$ such that $\begin{aligned} (3.9) & \int_{B_{σ} (y_{n})} | u_{n} |^{p} d x ⩾ \frac{μ}{2} . \end{aligned}$ Let ${\bar{u}}_{n} (x) = u_{n} (x + y_{n})$ , then by (3.9), we derive that $\begin{aligned} (3.10) & \int_{B_{σ} (0)} | {\bar{u}}_{n} |^{p} d x ⩾ \frac{μ}{2} . \end{aligned}$ Since ${u_{n}} \subset W$ is bounded and ${\bar{u}}_{n} (x) = u_{n} (x + y_{n})$ , then up to a subsequence, there exists $\bar{u} \in W$ such that $\begin{matrix} (3.11) & \{\begin{array}{ll} {\bar{u}}_{n} ⇀ \bar{u} & in W, \\ {\bar{u}}_{n} \to \bar{u} & in L_{loc}^{q} (R^{3}), p ⩽ q < p_{s}^{*}, \\ {\bar{u}}_{n} \to \bar{u} & a.e. x \in R^{3} . \end{array} \end{matrix}$ It follows from (3.10) and (3.11) that $\begin{array}{r} \int_{B_{σ} (0)} | \bar{u} |^{p} d x ⩾ \frac{μ}{2} . \end{array}$ This means that $\bar{u} \neq 0$ . In addition, since ${\bar{u}}_{n} ⇀ \bar{u}$ in W and $I^{'} (u_{n}) \to 0$ as $n \to + \infty$ , we prove that $I^{'} ({\bar{u}}_{n}) \to 0$ as $n \to + \infty$ . It follows from (3.11), Lemma 3.1, (iii) of Proposition 2.3 and Proposition 5.4.7 of [31] that $⟨ I^{'} (\bar{u}), v ⟩ = 0$ for every $v \in W$ . We finish the proof. □

By using subtle scaling transformation, from the following result, we see that the functional $I$ obtains the mountain pass geometry.

Lemma 3.3.

Suppose that $\frac{(2 t + 4 s) p}{2 t + p s} < q < p_{s}^{*}$ and $λ \in (0, + \infty)$ . Then

there is $\bar{v} \in W$ satisfying $I (\bar{v}) < 0$ ;

$\bar{c} : = {inf}_{γ \in Γ} {max}_{t \in [0, 1]} I (γ (t)) > 0$ , where $Γ = {γ \in C ([0, 1], W) : γ (0) = 0, I (γ (1)) < 0}$ .

Proof.

Let $\frac{(2 t + 4 s) p}{2 t + p s} < q < p_{s}^{*}$ and take $u \in W ∖ {0}$ . For $h > 0$ , denote $u_{h} (x) = h^{\frac{2 t + p s}{4 - p}} u (h x)$ . Then by (i) of Proposition 2.3 and direct calculations, we have $\begin{matrix} I (u_{h}) = \frac{h^{β_{1}}}{p} {[u]}_{s, p}^{p} + \frac{h^{β_{2}}}{p} ‖ u ‖_{p}^{p} + \frac{λ h^{β_{1}}}{4} {[ϕ_{u}^{t}]}_{t, 2}^{2} - \frac{h^{β_{3}}}{q} ‖ u ‖_{q}^{q}, \end{matrix}$ where $\begin{matrix} (3.12) & β_{1} = \frac{4 (2 t + p s)}{4 - p} - 2 t - 3, β_{2} = \frac{p (2 t + p s)}{4 - p} - 3, β_{3} = \frac{q (2 t + p s)}{4 - p} - 3 . \end{matrix}$ Since $\frac{(2 t + 4 s) p}{2 t + p s} < q$ and $\frac{12}{3 + 2 t + 4 s} < p < \frac{12}{3 + 2 t}$ , it holds that $β_{3} > β_{1} > β_{2}$ , $β_{3} > 0$ . Therefore there exists $h_{0} > 0$ such that $I (u_{h_{0}}) < 0$ .

By Lemma 2.1 and (2.7), we infer that $I (u) ⩾ \frac{1}{p} ‖ u ‖^{p} - C ‖ u ‖^{q}$ . Therefore there are $ρ > 0$ , $δ > 0$ satisfying $\begin{aligned} (3.13) & I (u) ⩾ δ, for any ‖ u ‖ = ρ . \end{aligned}$ Combining $I (0) = 0$ , (i) and (3.13), we prove that (ii) holds. We finish the proof.

□

Now we are ready to prove Theorem 1.1. In fact, combining Lemma 3.3 and the mountain pass lemma, we obtain the existence of (PS) sequences for $I$ . We should get the boundedness of the (PS) sequences of the functional $I$ . We are now in the case of $\frac{(2 t + 4 s) p}{2 t + p s} < q < p_{s}^{*}$ . We introduce the following auxiliary functional $\tilde{I} : W \to R$ with the numbers given in (3.12) $\begin{matrix} \tilde{I} (u) : = \frac{β_{1}}{p} {[u]}_{s, p}^{p} + \frac{β_{2}}{p} ‖ u ‖_{p}^{p} + \frac{λ β_{1}}{4} {[ϕ_{u}^{t}]}_{t, 2}^{2} - \frac{β_{3}}{q} ‖ u ‖_{q}^{q} . \end{matrix}$ Adopting the method in [15], by Theorem 2.8 of [30], a bounded (PS) sequence of functional $I$ at the level $\bar{c}$ is constructed as follows.

Lemma 3.4.

Suppose $\frac{(2 t + 4 s) p}{2 t + p s} < q < p_{s}^{*}$ . There exists a bounded sequence ${u_{n}} \subset W$ such that $\begin{aligned} (3.14) & I (u_{n}) \to \bar{c} > 0, I^{'} (u_{n}) \to 0 and \tilde{I} (u_{n}) \to 0, as n \to + \infty . \end{aligned}$

Proof.

Firstly, we define the map $G : R \times W \to W$ by $G (ς, u (x)) : = e^{\frac{2 t + p s}{4 - p} ς} u (e^{ς} x)$ . Then similar to Lemma 3.3, we get $\begin{aligned} (3.15) & (I \circ G) (ς, u) = \frac{e^{β_{1} ς}}{p} {[u]}_{s, p}^{p} + \frac{e^{β_{2} ς}}{p} ‖ u ‖_{p}^{p} + \frac{λ e^{β_{1} ς}}{4} {[ϕ_{u}^{t}]}_{t, 2}^{2} - \frac{e^{β_{3} ς}}{q} ‖ u ‖_{q}^{q} \end{aligned}$ It is obvious that $I \circ G$ is continuously Fréchet-differentiable on $R \times W$ . Since $β_{3} > β_{1} > β_{2}$ and $β_{3} > 0$ , we obtain that for each fixed $u \in W ∖ {0}$ , $\begin{aligned} (3.16) & (I \circ G) (ς, u) \to - \infty as ς \to + \infty . \end{aligned}$ We denote $\begin{matrix} \tilde{Γ} = {\tilde{γ} \in C ([0, 1], R \times W) : \tilde{γ} (0) = (0, 0), (I \circ G) (\tilde{γ} (1)) < 0} . \end{matrix}$ It follows from (3.16) and $(I \circ G) (0, 0) = 0$ that $\begin{aligned} (3.17) & \tilde{c} : = inf_{\tilde{γ} \in \tilde{Γ}} sup_{l \in [0, 1]} (I \circ G) (\tilde{γ} (l)) \end{aligned}$ is well defined. Next we will show that $\begin{aligned} (3.18) & \tilde{c} = \bar{c} > 0, \end{aligned}$ where Γ and $\bar{c}$ are defined by (ii) of Lemma 3.3.

In fact, first of all, we prove $\tilde{c} ⩽ \bar{c}$ . Since ${0} \times Γ \subset \tilde{Γ}$ , then we get $\tilde{c} ⩽ \bar{c}$ . Secondly, we claim $\bar{c} ⩽ \tilde{c}$ . Since ${G \circ \tilde{γ} : \tilde{γ} \in \tilde{Γ}} \subset Γ$ , hence we obtain $\bar{c} ⩽ \tilde{c}$ . Therefore, we conclude that $\tilde{c} = \bar{c} > 0$ . By the definition of $\bar{c}$ , there is a sequence ${γ_{n}} \subset Γ$ satisfying ${max}_{l \in [0, 1]} I (γ_{n} (l)) ⩽ \bar{c} + \frac{1}{n}$ . Then we obtain $\begin{aligned} (3.19) & sup_{l \in [0, 1]} (I \circ G) ({\tilde{γ}}_{n} (l)) ⩽ \bar{c} + \frac{1}{n}, \end{aligned}$ where ${\tilde{γ}}_{n} (l) = (0, γ_{n} (l)) \subset \tilde{Γ}$ . From (3.17)–(3.19), by Theorem 2.8 of [30], there exists a sequence ${(τ_{n}, ω_{n})} \subset R \times W$ so that as $n \to + \infty$ , $\begin{matrix} (3.20) & \{\begin{array}{l} (I \circ G) (τ_{n}, ω_{n}) \to \bar{c}, \\ {(I \circ G)}^{'} (τ_{n}, ω_{n}) \to 0, \\ {min}_{l \in [0, 1]} ‖ (τ_{n}, ω_{n}) - {\tilde{γ}}_{n} (l) ‖_{R \times W} \to 0, \end{array} \end{matrix}$ where $‖ (ς, u) ‖_{R \times W} = {(| ς |^{2} + ‖ u ‖^{2})}^{\frac{1}{2}}$ . In addition, we get $\begin{array}{r} (3.21) & ⟨ {(I \circ G)}^{'} (τ_{n}, ω_{n}), (ϑ, v) ⟩ = ⟨ I^{'} (G (τ_{n}, ω_{n})), G (τ_{n}, v) ⟩ + \tilde{I} (G (τ_{n}, ω_{n})) ϑ, \forall (ϑ, v) \in R \times W . \end{array}$ Substitute $(ϑ, v) = (1, 0)$ and $(ϑ, v) = (0, G (- τ_{n}, φ))$ into (3.21), respectively and suppose $u_{n} = G (τ_{n}, ω_{n})$ . From (3.20) and (3.21), we obtain that (3.14) holds. We infer that for n sufficiently large, $\begin{aligned} \bar{c} + 1 & ⩾ I (u_{n}) - \frac{1}{β_{3}} \tilde{I} (u_{n}) \\ = \frac{β_{3} - β_{1}}{p β_{3}} {[u_{n}]}_{s, p}^{p} + \frac{β_{3} - β_{2}}{p β_{3}} ‖ u_{n} ‖_{p}^{p} + \frac{λ (β_{3} - β_{1})}{4 β_{3}} {[ϕ_{u_{n}}^{t}]}_{t, 2}^{2} . \end{aligned}$ Thus ${u_{n}}$ is bounded in W. We finish the proof. □

Proof of Theorem 1.1.

By Lemma 3.4, there exists a bounded sequence ${u_{n}}$ in W such that $\begin{matrix} I (u_{n}) \to \bar{c} > 0, I^{'} (u_{n}) \to 0 and \tilde{I} (u_{n}) \to 0, as n \to + \infty . \end{matrix}$ From Lemma 3.2, there exists $u \in W ∖ {0}$ satisfying $I^{'} (u) = 0$ . Hence system (1.1) admits at least one non-trivial solution. We complete the proof. □

Next we will give a proof of Theorem 1.2. The main difficulty is to find the boundedness of (PS) sequences for $I$ . We emphasize that the approach applied in Lemma 3.4 is not effective for the case $p < q ⩽ \frac{(2 t + 4 s) p}{2 t + p s}$ . To overcome this difficulty, applying the method in [16], we define a cut-off functional $I_{K} : W \to R$ by $\begin{aligned} (3.22) & I_{K} (u) = \frac{1}{p} ‖ u ‖^{p} + \frac{λ}{4} Ψ_{K} (u) {[ϕ_{u}^{t}]}_{t, 2}^{2} - \frac{1}{q} ‖ u ‖_{q}^{q}, \end{aligned}$ where $\begin{aligned} (3.23) & Ψ_{K} (u) : = ζ (\frac{‖ u ‖^{2}}{K^{2}}), ζ \in C_{0}^{\infty} (R^{+}, [0, 1]) \end{aligned}$ satisfying $ζ (l) = 1$ for $0 ⩽ l ⩽ \frac{1}{2}$ and $supp ζ \subset [0, 1]$ . If $‖ u ‖^{2} < \frac{K^{2}}{2}$ , then it is easy to check that the critical point $u \in W$ of $I_{K} (u)$ must be the critical point of $I$ and $\begin{matrix} ⟨ I_{K}^{'} (u), u ⟩ = ‖ u ‖^{p} + λ (Ψ_{K} (u) + \frac{‖ u ‖^{2}}{2 K^{2}} ζ^{'} (\frac{‖ u ‖^{2}}{K^{2}})) {[ϕ_{u}^{t}]}_{t, 2}^{2} - ‖ u ‖_{q}^{q} . \end{matrix}$

For all fixed $K > 0$ , we now infer that the functional $I_{K}$ possesses a mountain pass geometry in the next lemma.

Lemma 3.5.

Suppose $p < q ⩽ \frac{(2 t + 4 s) p}{2 t + p s}$ . Then we have:

there is $\bar{v} \in W$ satisfying $I_{K} (\bar{v}) < 0$ ;

${\bar{c}}_{K} : = {inf}_{γ \in Γ_{K}} {max}_{l \in [0, 1]} I_{K} (γ (l)) > 0$ , where $Γ_{K} = {γ \in C ([0, 1], W) : γ (0) = 0, I_{K} (γ (1)) < 0}$ .

Proof.

For every fixed $u \in W ∖ {0}$ and $h > \frac{K}{‖ u ‖}$ , we obtain that $\begin{matrix} I_{K} (h u) = \frac{h^{p}}{p} ‖ u ‖^{p} + \frac{λ h^{4}}{4} ζ (\frac{h^{2} ‖ u ‖^{2}}{K^{2}}) {[ϕ_{u}^{t}]}_{t, 2}^{2} - \frac{h^{q}}{q} ‖ u ‖_{q}^{q} . \end{matrix}$ Taking $v = \tilde{h} u$ , where $\tilde{h} > \frac{K}{‖ u ‖}$ is sufficiently large, we check that (i) holds for $p < q$ .

By (2.7), (3.23) and Lemma 2.1, we get that $\begin{aligned} (3.24) & I_{K} (u) ⩾ \frac{1}{p} ‖ u ‖^{p} - C ‖ u ‖^{q} . \end{aligned}$ Similar to the proof of Lemma 3.3, we derive (ii). From $I (0) = 0$ , (i) and (3.24), we obtain that (ii) holds. The proof is complete.

□

Based on the mountain pass lemma, there exists ${u_{n}} \subset W$ such that $\begin{aligned} (3.25) & I_{K} (u_{n}) \to {\bar{c}}_{K} > 0 I_{K}^{'} (u_{n}) \to 0 as n \to + \infty . \end{aligned}$

Now we study the property of the sequence ${u_{n}}$ satisfying (3.25) in the following lemma.

Lemma 3.6.

Suppose $p < q ⩽ \frac{(2 t + 4 s) p}{2 t + p s}$ and ${u_{n}} \subset W$ satisfies ( 3.25 ). Then for $K > 0$ sufficiently large, there is $λ^{*} = λ^{*} (K) > 0$ so that for each $0 < λ < λ^{*}$ , $\begin{aligned} (3.26) & \underset{n \to + \infty}{lim sup} ‖ u_{n} ‖ < \frac{K}{2} . \end{aligned}$

Proof.

Firstly, we infer that ${u_{n}}$ is bounded. Otherwise, $‖ u_{n} ‖ \to + \infty$ as $n \to + \infty$ , then by (3.23), we obtain $\begin{matrix} Ψ_{K} (u_{n}) = ζ (\frac{‖ u_{n} ‖}{‖ K ‖}) = 0, for every large n \in N, \end{matrix}$ and hence for every $n \in N$ large, $\begin{matrix} I_{K} (u_{n}) = \frac{1}{p} ‖ u_{n} ‖^{p} - \frac{1}{q} ‖ u_{n} ‖_{q}^{q} \to {\bar{c}}_{K}, \end{matrix}$ which is impossible since $‖ u_{n} ‖ \to + \infty$ and $p < q$ . Furthermore, we get $\begin{aligned} (\frac{1}{p} - \frac{1}{q}) ‖ u_{n} ‖^{p} + \frac{1}{q} ⟨ I_{K}^{'} (u_{n}), u_{n} ⟩ \\ = I_{K} (u_{n}) + (\frac{λ}{4} - \frac{λ}{q}) Ψ_{K} (u_{n}) {[ϕ_{u_{n}}^{t}]}_{t, 2}^{2} + \frac{λ ‖ u_{n} ‖^{2}}{2 q K^{2}} ζ^{'} (\frac{‖ u_{n} ‖^{2}}{K^{2}}) {[ϕ_{u_{n}}^{t}]}_{t, 2}^{2} \\ (3.27) & ⩽ I_{K} (u_{n}) + | \frac{λ}{4} - \frac{λ}{q} | Ψ_{K} (u_{n}) {[ϕ_{u_{n}}^{t}]}_{t, 2}^{2} + \frac{λ ‖ u_{n} ‖^{2}}{2 q K^{2}} | ζ^{'} (\frac{‖ u_{n} ‖^{2}}{K^{2}}) | {[ϕ_{u_{n}}^{t}]}_{t, 2}^{2} . \end{aligned}$ Next we prove (3.26) indirectly. Suppose, up to a subsequence, that $\begin{aligned} (3.28) & lim_{n \to + \infty} ‖ u_{n} ‖ ⩾ \frac{K}{2} . \end{aligned}$ Since ${u_{n}}$ is bounded, then by (3.25) and (3.28), we get $\begin{aligned} (3.29) & (\frac{1}{p} - \frac{1}{q}) ‖ u_{n} ‖^{p} + \frac{1}{q} ⟨ I_{K}^{'} (u_{n}), u_{n} ⟩ ⩾ C K^{p} - \frac{1}{q} ‖ I_{K}^{'} (u_{n}) ‖ ‖ u_{n} ‖ ⩾ C K^{p} - K \end{aligned}$ for n sufficiently large. Combining (i) of Proposition 2.3 and $ζ \in C_{0}^{\infty} (R^{+}, [0, 1])$ , we have $\begin{aligned} (3.30) & Ψ_{K} (u_{n}) {[ϕ_{u_{n}}^{t}]}_{t, 2}^{2} ⩽ C ‖ u_{n} ‖^{4} Ψ_{K} (u_{n}) ⩽ C ‖ u_{n} ‖^{4} \end{aligned}$ and $\begin{aligned} (3.31) & ‖ u_{n} ‖^{2} | ζ^{'} (\frac{‖ u_{n} ‖^{2}}{K^{2}}) | {[ϕ_{u_{n}}^{t}]}_{t, 2}^{2} ⩽ C ‖ u_{n} ‖^{6} | ζ^{'} (\frac{‖ u_{n} ‖^{2}}{K^{2}}) | ⩽ C ‖ u_{n} ‖^{6} . \end{aligned}$

By Lemma 3.5 and (ii) of Proposition 2.3, there hold $\begin{aligned} {\bar{c}}_{K} ⩽ & max_{l \in [0, 1]} I_{K} (l \bar{v}) \\ (3.32) & ⩽ & max_{l \in [0, 1]} {\frac{l^{p}}{p} ‖ \bar{v} ‖^{p} - \frac{1}{q} ‖ \bar{v} ‖_{q}^{q}} + max_{l \in [0, 1]} {\frac{λ l^{4}}{4} Ψ_{K} (l \bar{v}) {[ϕ_{\bar{v}}^{t}]}_{t, 2}^{2}} . \end{aligned}$ Similar to (3.30), we conclude that $\begin{array}{r} max_{l \in [0, 1]} {\frac{λ l^{4}}{4} Ψ_{K} (l \bar{v}) {[ϕ_{\bar{v}}^{t}]}_{t, 2}^{2}} ⩽ max_{l \in [0, 1]} {\frac{λ l^{4}}{4} Ψ_{K} (l \bar{v}) ‖ \bar{v} ‖^{4}} ⩽ C λ K^{4} . \end{array}$ It together with (3.32) means that $\begin{matrix} {\bar{c}}_{K} ⩽ C + C λ K^{4} . \end{matrix}$ Combining with $I_{K} (u_{n}) \to {\bar{c}}_{K}$ as $n \to + \infty$ , we infer that for n sufficiently large $\begin{aligned} (3.33) & I_{K} (u_{n}) ⩽ C + C λ K^{4} . \end{aligned}$ From (3.30)–(3.33) and (3.27), we see that for n large enough, $\begin{aligned} (3.34) & (\frac{1}{p} - \frac{1}{q}) ‖ u_{n} ‖^{p} + \frac{1}{q} ⟨ I_{K}^{'} (u_{n}), u_{n} ⟩ ⩽ C + C λ K^{4} . \end{aligned}$ By (3.29) and (3.34), we get $\begin{aligned} (3.35) & C + C λ K^{4} ⩾ C K^{p} - K, \end{aligned}$ where $C > 0$ is independent of K and λ. For $K > 0$ sufficiently large and $0 < λ < K^{- 4}$ , the inequality (3.35) would not hold. We finish the proof. □

Proof of Theorem 1.2.

According to Lemma 3.5, for every fixed $K > 0$ and $λ > 0$ , $I_{K}$ possesses a mountain pass level ${\bar{c}}_{K} > 0$ and there exists ${u_{n}} \subset W$ such that (3.25) holds. From Lemma 3.6, we choose $K > 0$ sufficiently large and $λ^{*} = λ^{*} (K) > 0$ small such that for all $0 < λ < λ^{*}$ $\begin{matrix} \underset{n \to + \infty}{lim sup} ‖ u_{n} ‖ < \frac{K}{2} . \end{matrix}$ This together with (3.14) and (3.23), we see that for n sufficiently large, $I_{K} (u_{n}) = I (u_{n})$ and $I_{K}^{'} (u_{n}) = I^{'} (u_{n})$ . Hence we obtain $I (u_{n}) \to {\bar{c}}_{K} > 0$ and $I^{'} (u_{n}) \to 0$ as $n \to + \infty$ . According to Lemma 3.2, there is $\bar{u} \in W ∖ {0}$ satisfying $I^{'} (\bar{u}) = 0$ . We finish the proof. □

Footnotes

Acknowledgements

Chungen Liu was partially supported by the NSF of China (11790271, 12171108), Guangdong Basic and Applied basic Research Foundation (2020A1515011019).

Yuyou Zhong was partially supported by the Innovative Research Funding Program for Graduate Students of Guangzhou University (2022GDJC-D09).

Jiabin Zuo was supported by the Guangdong Basic and Applied Basic Research Foundation (2022A1515110907).

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Non-trivial solutions for the fractional Schrödinger–Poisson system with p -Laplacian

Abstract

Keywords

1. Introduction and main results

Lemma 2.1 ([23]).

Footnotes

Acknowledgements

References