Positive ground state solutions for a class of Schrödinger–Poisson systems in R 4 involving critical Sobolev exponent

Abstract

In this paper, we study the following Schrödinger–Poisson system $\begin{matrix} \{\begin{matrix} - Δ u + u + λ ϕ u = f (u) + u^{3}, & x \in R^{4}, \\ - Δ ϕ = u^{2}, & x \in R^{4}, \end{matrix} \end{matrix}$ where $λ > 0$ is a parameter, $f \in C (R)$ and the nonlinear growth of $u^{3}$ reaches the Sobolev critical exponent since $2^{*} = 4$ in dimension 4. Under some suitable assumptions on f, we establish the existence of a positive radial and non-radial ground state solutions for the above system by using variational methods. We also discuss the asymptotic behavior of solutions with respect to the parameter λ.

Keywords

Schrödinger–Poisson system variational method critical exponent ground state solution

1. Introduction and main results

In this paper, we consider the following nonlinear Schrödinger–Poisson system $\begin{matrix} (P_{λ}) & \{\begin{matrix} - Δ u + u + λ ϕ u = f (u) + | u |^{2} u, & x \in R^{4}, \\ - Δ ϕ = u^{2}, & x \in R^{4}, \end{matrix} \end{matrix}$ where $λ > 0$ is a parameter and $f : R \to R$ satisfy the following assumptions:

$f \in C (R)$ be such that $f (t) = 0$ for all $t ⩽ 0$ ;

${lim}_{t \to 0^{+}} \frac{f (t)}{t} = 0$ ;

${lim}_{t \to \infty} \frac{f (t)}{t^{3}} = 0$ ;

there exists $θ_{0} \in (0, 1)$ such that for all $t > 0$ and $τ > 0$ $\begin{matrix} [\frac{f (τ)}{τ^{3}} - \frac{f (t τ)}{{(t τ)}^{3}}] sign (1 - t) + θ_{0} \frac{| 1 - t^{2} |}{{(t τ)}^{2}} ⩾ 0; \end{matrix}$

there exist $C_{1} > 0$ and $q \in (2, 4)$ such that $\begin{matrix} F (t) ⩾ C_{1} | t |^{q}, \forall t ⩾ 0, \end{matrix}$ where (and in the sequel) $F (u) = \int_{0}^{u} f (s) d s$ .

The general form of the Schrödinger–Poisson system is

\begin{matrix} (1.1) & \{\begin{matrix} - Δ u + V (x) u + λ K (x) ϕ u = f (x, u), & x \in R^{N}, \\ - Δ ϕ = K (x) u^{2}, & x \in R^{N} . \end{matrix} \end{matrix}

System (1.1) is also called Schrödinger–Maxwell system, arises in an interesting physical context. In fact, according to a classical model, the interaction of a charge particle with an electromagnetic field can be described by coupling the nonlinear Schrödinger’s and Poisson’s equations. For more information on the physical relevence of the Schrödinger–Poisson system, we refer the readers to the papers [6,12,25] and the references therein.

System (1.1) with $λ = 0$ is the classical Schrödinger equation $\begin{matrix} (1.2) & - Δ u + V (x) u = f (x, u) in R^{N}, \end{matrix}$ that has been widely investigated by many authors, specially on the existence of ground state solutions, positive solutions, sign-changing solutions and multiplicity of standing wave solutions, we refer the interested readers to [1,7,13,14,21,22,28,29] and the references therein.

Since the pioneering work of Benci and Fortunato [5], there are huge literature on the studies of the existence and behavior of solutions for system (1.1) under various assumptions on V, K and f. When $N = 3$ , using variational methods in combination with critical points theory, a lot of interesting results on the existence of positive solutions, ground state solutions, sign-changing solutions, infinitely many small energy solutions and infinitely many high energy solutions for system (1.1) can be founded in [2,3,9–11,16,17,19,20,24,26,33,34].

Besides, since the pioneering work of Brézis and Nirenberg [8], the nonlinear elliptic problems involving critical Sobolev exponents have been attracted much more attention. Recently, Huang et al. [15] considered system (1.1) with $N = 3$ , $λ = 1$ , $V (x) \equiv 1$ and $f (x, u) = k (x) | u |^{4} u + μ h (x) u$ . By using variational methods, they obtained a positive and sign-changing solutions for system (1.1) under appropriate assumptions on μ, K, k and h. In [35], Zhang studied system (1.1) when $N = 3$ , $λ = 1$ , $V \equiv 1$ and $f (x, u) = a (x) | u |^{p - 2} u + u^{5}$ . So, under some suitable assumptions on K and a, positive ground state solutions as well as nodal solutions were obtained by means of variational methods. Sun and Ma [27] considered system (1.1) with $N = 3$ , $λ = 1$ and $K (x) \equiv 1$ . When the potential function is positive and 1-periodic in $x \in R^{3}$ , and the nonlinear term $f (x, u)$ is superlinear in u and satisfies the well known variant Nehari-type condition ((VN) for short), that is, $\begin{matrix} (1.3) & \frac{f (x, t)}{| t |^{3}} is increasing on (- \infty, 0) \cup (0, \infty), \forall x \in R^{3}, \end{matrix}$ the authors obtained a ground state solutions by means of Nehari manifold method. Furthermore, they also investigated system (1.1) in the critical case where $f (x, u) = K (x) | u |^{4} u + λ g (x, u)$ and g satisfies (VN)-condition. By the aid of Nehari manifold method, they proved the existence of ground state solutions. For more results about the Schrödinger–Poisson system with critical nonlinearities, we refer the readers to [18,23,32,36].

Inspired by the above works, in the present paper, by using a variant version of mountain pass lemma without the Palais–Smale condition, we obtain a new results on the existence of positive and radial positive ground state solutions for the system ( P λ ) setting on the whole space $R^{4}$ , which extend some recent results in the literature. Moreover, the asymptotic behavior of solutions with respect to the parameter λ is discussed.

Definition 1.1.
u is said to be a ground state of problem ( P λ ) if u is a solution of problem ( P λ ) and has the least energy among all the nontrivial solutions of problem ( P λ ).

The main results of this paper are the following theorems.
Theorem 1.2.
Assume that $(f_{0})$ – $(f_{4})$ hold. Then, there exists $λ^{} > 0$ such that for all* $λ \in (0, λ^{})$ , the system (* P λ ) has a positive non-radial ground state solution $(u_{λ}, ϕ_{λ}) \in H^{1} (R^{4}) \times D^{1, 2} (R^{4})$ .
Theorem 1.3.
Under the assumptions of Theorem 1.2 , there exists $λ^{} > 0$ such that for all* $λ \in (0, λ^{})$ , the system (* P λ ) admits a radial positive ground state solution $(u_{λ}, ϕ_{λ}) \in H_{r}^{1} (R^{4}) \times D^{1, 2} (R^{4})$ . Moreover, For any sequence ${λ_{n}}$ with $λ_{n} ↘ 0$ as $n \to \infty$ , there exists a subsequence, still denoted by ${λ_{n}}$ , such that $(u_{λ_{n}}, ϕ_{λ_{n}})$ converges to $(u_{0}, 0)$ strongly in $H^{1} (R^{4}) \times D^{1, 2} (R^{4})$ as $n \to \infty$ , where $u_{0}$ is a radial positive ground state solution to the limit problem $\begin{matrix} (P_{0}) & - Δ u + V (x) u = f (u) in R^{4} . \end{matrix}$
Remark 1.4.

Since the problem ( P λ ) is defined in the whole space $R^{4}$ and the nonlinearity involving the critical Sobolev exponent, the main difficulty of this problem is the lack of compactness for Sobolev embedding theorem.

To the best of our knowledge, condition $(f_{3})$ which is weaker than (VN) was first introduced by Tang and Cheng [30] to deal with the nonlinear Kirchhoff-type equation on bounded domain, and then by Chen and Tang [11] for deal with the nonlinear Schrödinger–Poisson system setting on $R^{3}$ .

The paper is organized as follows. In Section 2, we present some preliminary results and we establish the variational framework associated with problem ( P λ ). Section 3 is devoted to the proof of Theorems 1.2 and 1.3.
2. Preliminaries and technical lemmas

In the following, we will introduce the variational setting for problem ( P λ ). In the sequel, we denote by $| \cdot |_{p}$ the usual norm of the space $L^{p} (R^{4})$ , $c_{i}$ or C stand for different positive constants, $H^{*}$ denotes the dual space of H, and → (resp. ⇀) denotes the strong (resp. weak) convergence.

As usual, for $1 ⩽ p < + \infty$ , we let $\begin{matrix} ‖ u ‖_{p} : = {(\int_{R^{4}} | u |^{p} d x)}^{\frac{1}{p}} . \end{matrix}$

Let $\begin{matrix} H^{1} (R^{4}) = {u \in L^{2} (R^{4}) : \nabla u \in L^{2} (R^{4})}, \end{matrix}$ with the inner product and norm $\begin{matrix} ⟨ u, v ⟩ : = {⟨ u, v ⟩}_{H^{1}} = \int_{R^{4}} (\nabla u \nabla v + u v) d x, ‖ u ‖ : = ‖ u ‖_{H^{1}} = {⟨ u, u ⟩}_{H^{1}}^{\frac{1}{2}} . \end{matrix}$ Therefore, by the Sobolev embedding theorem [31, Theorem 1.8], $H^{1} (R^{4}) ↪ L^{s} (R^{4})$ is continuous for each $s \in [2, 4]$ , that is, there exist $μ_{s} > 0$ such that $\begin{matrix} (2.1) & | u |_{s} ⩽ μ_{s} ‖ u ‖, \forall u \in H^{1} (R^{4}), s \in [2, 4] . \end{matrix}$

In order to prove Theorem 1.2, following [4] (see also [31, Theorem 1.31]), write the elements of $R^{4} = R^{2} \times R^{2}$ as $(x_{1}, x_{2})$ , $x_{1}, x_{2} \in R^{2}$ . Next, we consider the action of $G_{2} : = O (2) \times O (2)$ on $H^{1} (R^{4})$ defined by $\begin{matrix} g u (x) : = u (g^{- 1} x) . \end{matrix}$ Let τ be the involution defined on $R^{4} = R^{2} \times R^{2}$ by $\begin{matrix} τ (x_{1}, x_{2}) : = (x_{2}, x_{1}) . \end{matrix}$ The action of $G : = {id, τ}$ on $\begin{matrix} Fix (G_{2}) : = {u \in H^{1} (R^{4}) : g u = u, \forall g \in G_{2}} \end{matrix}$ is given by $\begin{matrix} g u (x) : = \{\begin{matrix} u (x), & if g = id; \\ - u (g^{- 1} x), & if g = τ . \end{matrix} \end{matrix}$ Set $\begin{matrix} H : = Fix (G) = {u \in Fix (G_{2}) : g u = u \forall g \in G} . \end{matrix}$ Note that 0 is the only radial function of H for this case.

We also define the following subspace of $H^{1} (R^{4})$ $\begin{matrix} (2.2) & H_{r} : = H_{r}^{1} (R^{4}) = {u \in H^{1} (R^{4}) : u (x) = u (| x |)} . \end{matrix}$ Then, the embedding $H_{r} ↪ L^{p} (R^{4})$ is compact for $p \in (2, 4)$ (see [31, Corollary 1.26]).

Let $D^{1, 2} (R^{4})$ be the completion of $C_{0}^{\infty} (R^{4})$ with respect to the norm $\begin{matrix} ‖ u ‖_{D^{1, 2}}^{2} = \int_{R^{4}} | \nabla u |^{2} d x . \end{matrix}$ Then, the embedding $D^{1, 2} (R^{4}) ↪ L^{4} (R^{4})$ is continuous (see for instance [31, Theorem 1.8]). In what follows, S denotes the best Sobolev constant: $\begin{matrix} (2.3) & S : = inf_{u \in D^{1, 2} (R^{4}) ∖ {0}} \frac{\int_{R^{4}} | \nabla u |^{2} d x}{{(\int_{R^{4}} | u |^{4} d x)}^{1 / 2}} . \end{matrix}$

For every $u \in H^{1} (R^{4})$ , by the Lax–Milgram theorem, there exists a unique $ϕ_{u} \in D^{1, 2} (R^{4})$ such that $\begin{matrix} (2.4) & - Δ ϕ_{u} = u^{2} in R^{4} . \end{matrix}$ Furthermore, $ϕ_{u}$ has the following integral expression $\begin{matrix} (2.5) & ϕ_{u} (x) = \frac{1}{4 π^{2}} \int_{R^{4}} \frac{u^{2} (y)}{| x - y |^{2}} d y ⩾ 0 . \end{matrix}$ From (2.4), for any $u \in H^{1} (R^{4})$ , using the Hölder inequality we obtain $\begin{matrix} (2.6) & ‖ ϕ_{u} ‖_{D^{1, 2}}^{2} = \int_{R^{4}} ϕ_{u} u^{2} d x ⩽ ‖ ϕ_{u} ‖_{4} ‖ u ‖_{8 / 3}^{2} ⩽ C ‖ ϕ_{u} ‖_{D^{1, 2}} ‖ u ‖_{8 / 3}^{2}, \end{matrix}$ where $C > 0$ is a constant. It is clear that $8 / 3 \in (2, 4)$ , therefore it follows from (2.1) that $\begin{matrix} (2.7) & ‖ ϕ_{u} ‖_{D^{1, 2}} ⩽ C ‖ u ‖_{8 / 3}^{2} ⩽ C_{2} ‖ u ‖^{2} . \end{matrix}$ By (2.6), (2.7) and the Sobolev inequality, we obtain $\begin{matrix} (2.8) & \int_{R^{4}} ϕ_{u} u^{2} d x ⩽ C_{2} ‖ u ‖^{4} . \end{matrix}$ Moreover, $ϕ_{u}$ has the following properties (for a proof, see [9,24])

Lemma 2.1.
For $u \in H^{1} (R^{4})$ we have
$ϕ_{t u} = t^{2} ϕ_{u}$ , for all $t ⩾ 0$ ;

if $u_{n} ⇀ u$ in $H^{1} (R^{4})$ , then $ϕ_{u_{n}} ⇀ ϕ_{u}$ in $D^{1, 2} (R^{4})$ and $\begin{matrix} \int_{R^{4}} ϕ_{u} u^{2} d x ⩽ \underset{n \to \infty}{lim inf} \int_{R^{4}} ϕ_{u_{n}} u_{n}^{2} d x; \end{matrix}$

if $u_{n} ⇀ u$ in $H_{r}$ , then $ϕ_{u_{n}} ⇀ ϕ_{u}$ in $D^{1, 2} (R^{4})$ and $\begin{matrix} lim_{n \to \infty} \int_{R^{4}} ϕ_{u_{n}} u_{n}^{2} d x = \int_{R^{4}} ϕ_{u} u^{2} d x; \end{matrix}$

if u is a radial function (i.e., $u (x) = u (| x |)$ ), then $ϕ_{u}$ is radial.

Now, we define the energy functional $J_{λ} : H^{1} (R^{4}) \to R$ associated with problem ( P λ ) by $\begin{matrix} (2.9) & J_{λ} (u) = \frac{1}{2} \int_{R^{4}} (| \nabla u |^{2} + | u |^{2}) d x + \frac{λ}{4} \int_{R^{4}} ϕ_{u} u^{2} d x - \int_{R^{4}} F (u) d x - \frac{1}{4} \int_{R^{4}} | u |^{4} d x . \end{matrix}$ Therefore, under the assumptions $(f_{1})$ – $(f_{2})$ , J is well defined and $J \in C^{1} (H^{1} (R^{4}), R)$ with $\begin{matrix} (2.10) & \begin{matrix} ⟨ J_{λ}^{'} (u), v ⟩ = & \int_{R^{4}} (\nabla u \nabla v + u v) d x + λ \int_{R^{4}} ϕ_{u} u v d x \\ - \int_{R^{4}} f (u) v d x - \int_{R^{4}} | u |^{2} u v d x, \forall v \in H^{1} (R^{4}) . \end{matrix} \end{matrix}$ Clearly, if $u \in H^{1} (R^{4})$ is a critical point of $J_{λ}$ , then the pair $(u, ϕ_{u})$ is a solution to system ( P λ ). Furthermore, if u is a critical point of $J_{λ}$ restricted to H then u is a critical point of $J_{λ}$ thanks to the principle of symmetric criticality (see [31, Theorem 1.28]).

We define the Nehari manifold associated to problem ( P λ ) by $\begin{matrix} (2.11) & N_{λ} = {u \in H ∖ {0} : ⟨ J_{λ}^{'} (u), u ⟩ = 0} . \end{matrix}$ Consequently, if $u \neq 0$ is a critical point of $J_{λ}$ (i.e., $J_{λ}^{'} (u) = 0$ ), then $u \in N_{λ}$ . Set $\begin{matrix} (2.12) & c_{λ} = inf_{u \in N_{λ}} J_{λ} (u) . \end{matrix}$

Similarly, the energy functional $J_{0} : H \to R$ associated with problem ( P 0 ) is given by $\begin{matrix} (2.13) & J_{0} (u) = \frac{1}{2} ‖ u ‖^{2} - \int_{R^{4}} F (u) d x - \frac{1}{4} \int_{R^{4}} | u |^{4} d x . \end{matrix}$ The Nehari manifold associated to problem ( P 0 ) is given by $\begin{matrix} (2.14) & N_{0} = {u \in H ∖ {0} : ⟨ J_{0}^{'} (u), u ⟩ = 0} . \end{matrix}$

In what follows, we shall prove that $c_{λ}$ is attained at some $u_{λ} \in N_{λ}$ which is a critical point of $J_{λ}$ , that is, $u_{λ}$ is a ground state solution of ( P λ ).
Lemma 2.2.
Assume that $(f_{3})$ holds. Then we have
$\begin{matrix} \frac{1 - t^{4}}{4} τ f (τ) + F (t τ) - F (τ) + \frac{θ_{0}}{4} {(1 - t^{2})}^{2} τ^{2} ⩾ 0, \forall t > 0, τ > 0 . \end{matrix}$ Particularly, if $t = 0$ we have $\begin{matrix} \frac{1}{4} τ f (τ) - F (τ) + \frac{θ_{0}}{4} τ^{2} ⩾ 0 . \end{matrix}$

$\begin{matrix} J_{λ} (u) ⩾ J_{λ} (t u) + \frac{1 - t^{4}}{4} ⟨ J_{λ}^{'} (u), u ⟩ + \frac{(1 - θ_{0}) {(1 - t^{2})}^{2}}{4} ‖ u ‖^{2}, \forall u \in H, t ⩾ 0 . \end{matrix}$

For any $u ⩾ 0$ , the function $h_{u} (s) : = \frac{1}{4} f (s u) s u - F (s u) + \frac{θ_{0}}{4} {(s u)}^{2}$ is increasing for $s > 0$ .

Proof.
The proof of conclusion (i) and (ii) is similar to the one of [11, Lemma 2.1 and Lemma 4.1]. So, we only need to prove the conclusion (iii). We have two cases need to be considered: $s ⩽ 1$ and $s ⩾ 1$ .

If $s ⩽ 1$ , using $(f_{3})$ and (i) one has $\begin{matrix} (2.15) & \begin{matrix} h_{u} (1) - h_{u} (s) & = \frac{1}{4} (f (u) u - f (s u) s u) + F (s u) - F (u) + \frac{θ_{0}}{4} (1 - s^{2}) u^{2} \\ ⩾ \frac{1}{4} (f (u) u - f (s u) s u) - \frac{1 - s^{4}}{4} f (u) u - \frac{θ_{0}}{4} {(1 - s^{2})}^{2} u^{2} + \frac{θ_{0}}{4} (1 - s^{2}) u^{2} \\ = \frac{s^{4}}{4} f (u) u - \frac{1}{4} f (s u) s u + \frac{θ_{0}}{4} (1 - s^{2}) {(s u)}^{2} \\ = \frac{1}{4} [\frac{f (u)}{u^{3}} - \frac{f (s u)}{{(s u)}^{3}} + θ_{0} \frac{1 - s^{2}}{{(s u)}^{2}}] {(s u)}^{4} \\ ⩾ 0 . \end{matrix} \end{matrix}$

If $s ⩾ 1$ , taking $τ = s u$ and $t = 1 / s$ in conclusion (i), we then get $\begin{matrix} \frac{s^{4} - 1}{4 s^{4}} f (s u) s u + F (u) - F (s u) + \frac{θ_{0}}{4 s^{2}} {(s^{2} - 1)}^{2} u^{2} ⩾ 0 . \end{matrix}$ Therefore, combining the last inequality with $(f_{3})$ we have $\begin{matrix} h_{u} (s) - h_{u} (1) & = \frac{1}{4} (f (s u) s u - f (u) u) + F (u) - F (s u) + \frac{θ_{0}}{4} (s^{2} - 1) u^{2} \\ ⩾ \frac{1}{4} (f (s u) s u - f (u) u) - \frac{s^{4} - 1}{4 s^{4}} f (s u) s u - \frac{θ_{0}}{4 s^{2}} {(s^{2} - 1)}^{2} u^{2} + \frac{θ_{0}}{4} (s^{2} - 1) u^{2} \\ = \frac{1}{4 s^{4}} f (s u) s u - \frac{1}{4} f (u) u + \frac{θ_{0}}{4 s^{2}} (s^{2} - 1) u^{2} \\ = \frac{1}{4} [\frac{f (s u)}{{(s u)}^{3}} - \frac{f (u)}{u^{3}} + θ_{0} \frac{s^{2} - 1}{{(s u)}^{2}}] u^{4} \\ ⩾ 0 . \end{matrix}$ This together with (2.15) implies that $h_{u} (t)$ is increasing for $t > 0$ . Hence, the proof is completed. □
Definition 2.3.

A sequence ${u_{n}} \subset H$ is said to be a Palais–Smale sequence at the level $c \in R$ ( ${(PS)}_{c}$ sequence for short) if $J (u_{n}) \to c$ and $J^{'} (u_{n}) \to 0$ .

The functional J is said to satisfy the Palais–Smale condition at the level c ( ${(PS)}_{c}$ condition for short) if any ${(PS)}_{c}$ sequence has a convergent subsequence.

In order to prove Theorem 1.2, we shall use the following version of mountain pass theorem without (PS) condition.
Proposition 2.4 ([31, Theorem 2.9]).

Let X be a Banach space, $J \in C^{1} (X, R)$ , $e \in X$ and $r > 0$ be such that $‖ e ‖ > r$ and $\begin{matrix} α : = inf_{‖ e ‖ = r} J (u) > J (0) ⩾ J (e) . \end{matrix}$ Define $\begin{matrix} Γ : = {γ \in C ([0, 1], X) | γ (0) = 0, γ (1) = e} . \end{matrix}$ If $\begin{matrix} c : = inf_{γ \in Γ} max_{t \in [0, 1]} J (γ (t)) ⩾ α, \end{matrix}$ then there exist a sequence ${u_{n}} \subset X$ satisfying $\begin{matrix} J (u_{n}) \to c and J^{'} (u_{n}) \to 0 . \end{matrix}$

Now, we show that the functional $J_{λ}$ has the mountain pass geometry.

Lemma 2.5.
Under the assumption of Theorem 1.2 , the functional $J_{λ}$ has the following properties:
There exist two constants $α, r > 0$ such that $\begin{matrix} J_{λ} (u) ⩾ α, \forall u \in H : ‖ u ‖ = r . \end{matrix}$

There exist $λ^{} > 0$ and* $e \in H$ such that $‖ e ‖ > r$ and $J_{λ} (e) ⩽ 0$ , $\forall λ \in (0, λ^{})$ .

Proof.
(I) By $(f_{1})$ and $(f_{2})$ , for any $ε > 0$ and $p \in (2, 4)$ , there exists $C_{ε} >$ such that $\begin{matrix} (2.16) & | f (u) | ⩽ ε (| u | + | u |^{3}) + C_{ε} | u |^{p - 1} \end{matrix}$ and $\begin{matrix} (2.17) & | F (u) | ⩽ ε (\frac{1}{2} | u |^{2} + \frac{1}{4} | u |^{4}) + \frac{C_{ε}}{p} | u |^{p}, \forall u \in R . \end{matrix}$ Therefore, taking $ε ⩽ \frac{1}{2 μ_{2}^{2}}$ , it follows from (2.1), (2.5), (2.9) and (2.17) that $\begin{matrix} J_{λ} (u) & = \frac{1}{2} ‖ u ‖^{2} + \frac{λ}{4} \int_{R^{4}} ϕ_{u} u^{2} d x - \int_{R^{4}} F (u) d x - \frac{1}{4} \int_{R^{4}} | u |^{4} d x \\ ⩾ \frac{1}{2} ‖ u ‖^{2} - \int_{R^{4}} F (u) d x - \frac{1}{4} \int_{R^{4}} | u |^{4} d x \\ ⩾ \frac{1}{2} ‖ u ‖^{2} - \frac{ε}{2} | u |_{2}^{2} - \frac{ε}{4} | u |_{4}^{4} - \frac{C_{ε}}{p} | u |_{p}^{p} - \frac{μ_{4}^{4}}{4} ‖ u ‖^{4} \\ ⩾ \frac{1}{2} ‖ u ‖^{2} - \frac{ε}{2} μ_{2}^{2} ‖ u ‖^{2} - \frac{C_{ε}}{p} μ_{p}^{p} ‖ u ‖^{p} - \frac{μ_{4}^{4}}{4} (1 + ε) ‖ u ‖^{4} \\ ⩾ \frac{1}{4} ‖ u ‖^{2} - c_{1} ‖ u ‖^{p} - c_{2} ‖ u ‖^{4} \end{matrix}$ for all $u \in H$ . Since $p > 2$ , choosing $r > 0$ sufficiently small, we conclude that there exists $α > 0$ such that $\begin{matrix} J_{λ} (u) ⩾ α, \forall u \in H, ‖ u ‖ = r . \end{matrix}$

(II) Let $v \in H$ be such that $| v |_{4} = 1$ . Then, by $(f_{4})$ and (2.9) we have $\begin{matrix} J_{λ} (t v) & = \frac{t^{2}}{2} ‖ v ‖^{2} + λ \frac{t^{4}}{4} \int_{R^{4}} ϕ_{v} v^{2} d x - \int_{R^{4}} F (t v) d x - \frac{t^{4}}{4} \int_{R^{4}} | v |^{4} d x \\ ⩽ \frac{t^{2}}{2} ‖ v ‖^{2} - C_{1} t^{q} | v |_{q}^{q} - \frac{t^{4}}{4} (1 - λ \int_{R^{4}} ϕ_{v} v^{2} d x) . \end{matrix}$ Therefore, choosing $λ^{} > 0$ such that $1 - λ^{} \int_{R^{4}} ϕ_{v} v^{2} d x > 0$ , we conclude from the preceding inequality that $\begin{matrix} lim_{t \to + \infty} J_{λ} (t v) \to - \infty, \forall λ \in (0, λ^{}) . \end{matrix}$ Thus, there exists $T > 0$ such that $‖ T v ‖ > r$ and $J_{λ} (T v) ⩽ 0$ , $\forall λ \in (0, λ^{})$ . The conclusion (II) follows by considering $e = T v$ . □
Lemma 2.6.
Assume that* $(f_{0})$ – $(f_{4})$ hold. Then, for each $u \in H ∖ {0}$ and $λ \in (0, λ^{})$ , there exists a unique* $t_{u} = t (u) > 0$ such that $t_{u} u \in N_{λ}$ and $J_{λ} (t_{u} u) = {max}_{t ⩾ 0} J_{λ} (t u)$ .
Proof.
Let $u \in H$ be fixed and define the map $h : [0, \infty) \to R$ by $\begin{matrix} h (t) = J_{λ} (t u) = \frac{t^{2}}{2} ‖ u ‖^{2} + λ \frac{t^{4}}{4} \int_{R^{4}} ϕ_{u} u^{2} d x - \int_{R^{4}} F (t u) d x - \frac{t^{4}}{4} \int_{R^{4}} | u |^{4} d x . \end{matrix}$ By Lemma 2.5, there exist $t = t_{u} > 0$ such that such that $h (t_{u}) = {max}_{t > 0} h (t)$ so that $h^{'} (t_{u}) = 0$ . Thus, $t_{u} u \in N_{λ}$ .

In the sequel, we claim that $t_{u}$ is unique for any $u \in H$ . To this end, arguing by contradiction, for any given $u \in H ∖ {0}$ , suppose that there exist $t_{1}, t_{2} > 0$ , $t_{1} \neq t_{2}$ such that $h^{'} (t_{1}) = h^{'} (t_{2}) = 0$ . Without loss of generality, we may assume that $t_{1} < t_{2}$ , then we have $\begin{matrix} (2.18) & \frac{1}{t_{1}^{2}} ‖ u ‖^{2} + λ \int_{R^{4}} ϕ_{u} u^{2} d x = \int_{R^{4}} \frac{f (t_{1} u)}{{(t_{1} u)}^{3}} u^{4} d x + \int_{R^{4}} | u |^{4} d x \end{matrix}$ and $\begin{matrix} (2.19) & \frac{1}{t_{2}^{2}} ‖ u ‖^{2} + λ \int_{R^{4}} ϕ_{u} u^{2} d x = \int_{R^{4}} \frac{f (t_{2} u)}{{(t_{2} u)}^{3}} u^{4} d x + \int_{R^{4}} | u |^{4} d x . \end{matrix}$ Since $t_{1} < t_{2}$ , it follows from $(f_{3})$ that $\begin{matrix} (2.20) & [\frac{f (t_{2} u)}{{(t_{2} u)}^{3}} - \frac{f (t_{1} u)}{{(t_{1} u)}^{3}}] u^{4} + \frac{t_{2}^{2} - t_{1}^{2}}{t_{2}^{2} t_{1}^{2}} θ_{0} u^{2} ⩾ 0 . \end{matrix}$ Therefore, combining (2.18), (2.19) and (2.20), one has $\begin{matrix} (\frac{1}{t_{2}^{2}} - \frac{1}{t_{1}^{2}}) ‖ u ‖^{2} & = \int_{R^{4}} [\frac{f (t_{2} u)}{{(t_{2} u)}^{3}} - \frac{f (t_{1} u)}{{(t_{1} u)}^{3}}] u^{4} d x \\ ⩾ (\frac{1}{t_{2}^{2}} - \frac{1}{t_{1}^{2}}) θ_{0} \int_{R^{4}} u^{2} d x, \end{matrix}$ which implies that $\begin{matrix} (\frac{1}{t_{2}^{2}} - \frac{1}{t_{1}^{2}}) [\int_{R^{4}} | \nabla u |^{2} d x + (1 - θ_{0}) \int_{R^{4}} u^{2} d x] ⩾ 0 . \end{matrix}$ This is an obvious contradiction since $t_{1} < t_{2}$ and $θ_{0} \in (0, 1)$ . Thus, $t_{1} = t_{2}$ . □
Remark 2.7.
We note that the minimum of $J_{λ}$ over $N_{λ}$ has the following characterization (see [31, Theorem 4.2]): $\begin{matrix} c_{λ} = inf_{u \in H ∖ {0}} max_{t ⩾ 0} J_{λ} (t u) = inf_{γ \in Γ} max_{t \in [0, 1]} J_{λ} (γ (t)), \end{matrix}$ where Γ is defined in Proposition 2.4.

3. Proof of main results

In this section, in order to prove Theorem 1.2, we shall investigate the minimizing sequence of $J_{λ}$ . We need the following compactness result due to P.L. Lions, see [31, Lemma 1.21].

Lemma 3.1.

Let $r > 0$ . If ${u_{n}}$ is bounded in $H^{1} (R^{4})$ and $\begin{matrix} lim_{n \to \infty} sup_{y \in R^{4}} \int_{B_{r} (y)} | u_{n} |^{2} d x = 0, \end{matrix}$ then we have $u_{n} \to 0$ in $L^{s} (R^{4})$ for any $s \in (2, 4)$ .

For all $ε > 0$ , we consider $\begin{matrix} (3.1) & U_{ε} (x) = \frac{2 \sqrt{2} ε}{ε^{2} + | x |^{2}}, x \in R^{4}, \end{matrix}$ which is the solution of the critical problem $- Δ u = u^{3}$ in $R^{4}$ and $\int_{R^{4}} | \nabla U_{ε} |^{2} d x = \int_{R^{4}} | U_{ε} |^{4} d x = S^{2}$ . Let $u_{ε} (x) = φ (x) U_{ε} (x)$ , where $φ \in C_{0}^{\infty} (R^{4}, [0, 1])$ is such that $φ (x) = 1$ for $| x | ⩽ R$ and $φ (x) = 0$ for $| x | ⩾ 2 R$ . Then, we have the following estimations as $ε \to 0^{+}$ (see [8]) $\begin{matrix} (3.2) & \int_{R^{4}} | \nabla u_{ε} |^{2} d x = S^{2} + O (ε^{2}), \int_{R^{4}} | u_{ε} |^{4} d x = S^{2} + O (ε^{4}), \end{matrix}$ and $\begin{matrix} (3.3) & \int_{R^{4}} | u_{ε} |^{s} d x = \{\begin{matrix} O (ε^{2} | ln ε |), & if s = 2, \\ O (ε^{4 - s}), & if s \in (2, 4) . \end{matrix} \end{matrix}$

Lemma 3.2.

$c_{λ} < \frac{1}{4} S^{2}$ , where $c_{λ}$ is defined by ( 2.12 ).

Proof.

For any $ε > 0$ , $t ⩾ 0$ , set $\begin{matrix} g (t) : = \frac{t^{2}}{2} \int_{R^{4}} | \nabla u_{ε} |^{2} d x - \frac{t^{4}}{4} \int_{R^{4}} | u_{ε} |^{4} d x . \end{matrix}$ Therefore, it is easy to check that g attains its maximum and $\begin{matrix} (3.4) & \begin{matrix} max_{t ⩾ 0} g (t) & = \frac{{(\int | \nabla u_{ε} |^{2})}^{2}}{4 \int | u_{ε} |^{4}} \\ = \frac{1}{4} S^{2} + O (ε^{2}) . \end{matrix} \end{matrix}$ On the other hand, by Lemma 2.6, we know that for any $ε > 0$ , there exists $t_{ε} > 0$ such that $t_{ε} u_{ε} \in N_{λ}$ and $J_{λ} (t_{ε} u_{ε}) = {max}_{t ⩾ 0} J_{λ} (t u_{ε})$ . Consequently, for $ε > 0$ sufficiently small, it follows from $(f_{4})$ , (2.6), (3.2), (3.3) and (3.4) that $\begin{matrix} (3.5) & \begin{matrix} c_{λ} & ⩽ J_{λ} (t_{ε} u_{ε}) = g (t_{ε}) + \frac{t_{ε}^{2}}{2} | u_{ε} |_{2}^{2} + \frac{λ t_{ε}^{4}}{4} \int_{R^{4}} ϕ_{u_{ε}} u_{ε}^{2} d x - \int_{R^{4}} F (t_{ε} u_{ε}) d x \\ ⩽ \frac{1}{4} S^{2} + O (ε^{2}) + O (ε^{2} | ln ε |) + C^{2} | u_{ε} |_{8 / 3}^{4} - C_{1} | u_{ε} |_{q}^{q} \\ ⩽ \frac{1}{4} S^{2} + O (ε^{2}) + O (ε^{2} | ln ε |) - O (ε^{4 - q}) . \end{matrix} \end{matrix}$ Then, the conclusion follows immediately from (3.5) since $4 - q < 2$ . □

Lemma 3.3.

Let $c \in (0, \frac{1}{4} S^{2})$ and ${u_{n}} \subset H$ be a bounded ${(PS)}_{c}$ sequence for $J_{λ}$ , then there exist $r, δ > 0$ and a sequence ${y_{n}} \in R^{4}$ such that $\begin{matrix} \underset{n \to \infty}{lim inf} \int_{B_{r} (y_{n})} | u_{n} |^{2} d x ⩾ δ > 0 . \end{matrix}$

Proof.

Arguing by contradiction, suppose that ${u_{n}}$ is vanishing, then Lemma 3.1 implies that $u_{n} \to 0$ in $L^{s} (R^{4})$ for all $s \in (2, 4)$ , therefore, by (2.16) and(2.17) we have $\begin{matrix} \int_{R^{4}} f (u_{n}) u_{n} d x = o (1) and \int_{R^{4}} F (u_{n}) d x = o (1), \end{matrix}$ where $o (1) \to 0$ as $n \to \infty$ . Moreover, it follows from (2.6) that $\begin{matrix} \int_{R^{4}} ϕ_{u_{n}} u_{n}^{2} d x = o (1) . \end{matrix}$ Since ${u_{n}}$ is a bounded ${(PS)}_{c}$ sequence of $J_{λ}$ , one has $\begin{matrix} (3.6) & c + o (1) = J_{λ} (u_{n}) = \frac{1}{2} ‖ u_{n} ‖^{2} - \frac{1}{4} \int_{R^{4}} | u_{n} |^{4} d x + o (1) \end{matrix}$ and $\begin{matrix} (3.7) & o (1) = ⟨ J_{λ}^{'} (u_{n}), u_{n} ⟩ = ‖ u_{n} ‖^{2} - \int_{R^{4}} | u_{n} |^{4} d x + o (1) . \end{matrix}$ Hence, we may assume that there is $b ⩾ 0$ such that $\begin{matrix} ‖ u_{n} ‖^{2} \to b, \int_{R^{4}} | u_{n} |^{4} d x \to b as n \to \infty . \end{matrix}$ By (2.3) we have $\begin{matrix} \int_{R^{4}} | \nabla u_{n} |^{2} d x ⩾ S {(\int_{R^{4}} | u_{n} |^{4} d x)}^{1 / 2}, \end{matrix}$ which implies that $\begin{matrix} (3.8) & b ⩾ S b^{1 / 2} . \end{matrix}$ Since $c > 0$ , then by (3.8) we have $b > 0$ . Therefore, combining (3.6) and (3.8) we have $\begin{matrix} (3.9) & c = \frac{1}{4} b ⩾ \frac{1}{4} S^{2}, \end{matrix}$ which is a contradiction with $c < \frac{1}{4} S^{2}$ . Thus ${u_{n}}$ can not be vanishing. The proof is completed. □

Lemma 3.4.

Suppose that ${u_{n}} \subset H$ be a ${(PS)}_{c}$ sequence for $J_{λ}$ with $λ \in (0, λ^{*})$ , then there exists $u_{λ} \in H$ such that $J_{λ}^{'} (u_{λ}) = 0$ . Furthermore, if $u_{λ} \neq 0$ , then passing to a subsequence, it holds that $\begin{matrix} \int_{R^{4}} ϕ_{u_{n}} u_{n}^{2} d x \to \int_{R^{4}} ϕ_{u_{λ}} u_{λ}^{2} d x . \end{matrix}$

Proof.

By Lemma 2.2(i), we have that $\begin{matrix} (3.10) & \begin{matrix} 1 + c + ‖ u_{n} ‖ & ⩾ J_{λ} (u_{n}) - \frac{1}{4} ⟨ J_{λ}^{'} (u_{n}), u_{n} ⟩ \\ = \frac{1}{4} ‖ u_{n} ‖^{2} + \int_{R^{4}} [\frac{1}{4} f (u_{n}) u_{n} - F (u_{n})] d x \\ ⩾ \frac{1}{4} \int_{R^{4}} | \nabla u_{n} |^{2} d x + \frac{1 - θ_{0}}{4} \int_{R^{4}} u_{n}^{2} d x \\ ⩾ \frac{1 - θ_{0}}{4} ‖ u_{n} ‖^{2}, \end{matrix} \end{matrix}$ hence, ${u_{n}}$ is bounded in H. So, going if necessary to a subsequence, we may assume that there exist $u_{λ} \in H$ and $B ⩾ 0$ such that $\begin{matrix} (3.11) & u_{n} ⇀ u_{λ} in H \end{matrix}$ and $\begin{matrix} \int_{R^{4}} ϕ_{u_{n}} u_{n}^{2} d x \to B . \end{matrix}$ If $u_{λ} = 0$ , the proof is complete. If $u_{λ} \neq 0$ , then, by Lemma 2.1(2) we have $\begin{matrix} \int_{R^{4}} ϕ_{u_{λ}} u_{λ}^{2} d x ⩽ B . \end{matrix}$ By virtue of $J_{λ}^{'} (u_{n}) \to 0$ in $H^{*}$ and (3.11) one has $\begin{matrix} \int_{R^{4}} (\nabla u_{λ} \nabla v + u_{λ} v) d x + λ \int_{R^{4}} ϕ_{u_{λ}} u_{λ} v d x - \int_{R^{4}} f (u_{λ}) v d x - \int_{R^{4}} | u_{λ} |^{2} u_{λ} v d x = 0 . \end{matrix}$ Therefore, if $\int_{R^{4}} ϕ_{u_{λ}} u_{λ}^{2} d x < B$ , then $⟨ J_{λ}^{'} (u_{λ}), u_{λ} ⟩ < 0$ . Hence, by Lemma 2.6 there exist a unique $t^{*} \in (0, 1]$ such that $\begin{matrix} ⟨ J_{λ}^{'} (t^{*} u_{λ}), t^{*} u_{λ} ⟩ = 0 and J_{λ} (t^{*} u_{λ}) = max_{t ⩾ 0} J_{λ} (t u_{λ}) > 0 . \end{matrix}$ By Lemma 2.2(iii), Fatou’s Lemma and the lower semicontinuity of the norm we have $\begin{matrix} c & ⩽ J_{λ} (t^{*} u_{λ}) = J_{λ} (t^{*} u_{λ}) - \frac{1}{4} ⟨ J_{λ}^{'} (t^{*} u_{λ}), t^{*} u_{λ} ⟩ \\ = \frac{{(t^{*})}^{2}}{4} ‖ u_{λ} ‖^{2} + \int_{R^{4}} [\frac{1}{4} f (t^{*} u_{λ}) t^{*} u_{λ} - F (t^{*} u_{λ})] d x \\ = \frac{{(t^{*})}^{2}}{4} \int_{R^{4}} | \nabla u_{λ} |^{2} d x + \frac{{(t^{*})}^{2} (1 - θ_{0})}{4} \int_{R^{4}} | u_{λ} |^{2} d x \\ + \int_{R^{4}} [\frac{1}{4} f (t^{*} u_{λ}) t^{*} u_{λ} - F (t^{*} u_{λ}) + \frac{θ_{0}}{4} {(t^{*} u_{λ})}^{2}] d x \\ ⩽ \frac{1}{4} \int_{R^{4}} | \nabla u_{λ} |^{2} d x + \frac{1 - θ_{0}}{4} \int_{R^{4}} | u_{λ} |^{2} d x + \int_{R^{4}} [\frac{1}{4} f (u_{λ}) u_{λ} - F (u_{λ}) + \frac{θ_{0}}{4} u_{λ}^{2}] d x \\ ⩽ \underset{n \to \infty}{lim inf} [\frac{1}{4} \int_{R^{4}} | \nabla u_{n} |^{2} d x + \frac{1 - θ_{0}}{4} \int_{R^{4}} | u_{n} |^{2} d x + \int_{R^{4}} (\frac{1}{4} f (u_{n}) u_{n} - F (u_{n}) + \frac{θ_{0}}{4} u_{n}^{2}) d x] \\ = \underset{n \to \infty}{lim inf} (J_{λ} (u_{n}) - \frac{1}{4} ⟨ J_{λ}^{'} (u_{n}), u_{n} ⟩) \\ = c . \end{matrix}$ Hence, we conclude that $t^{*} = 1$ and then $⟨ J_{λ}^{'} (u_{λ}), u_{λ} ⟩ = 0$ , this is a contradiction. Thus, $\begin{matrix} \int_{R^{4}} ϕ_{u_{n}} u_{n}^{2} d x \to \int_{R^{4}} ϕ_{u_{λ}} u_{λ}^{2} d x . \end{matrix}$ □

Proof of Theorem 1.2.

By Lemma 2.5, all conditions of Proposition 2.4 are satisfied. Hence, for all $λ \in (0, λ^{*})$ , there exists a bounded ${(PS)}_{c_{λ}}$ sequence ${v_{n}} \subset H$ for $J_{λ}$ . By Lemmas 3.2 and 3.3, there exist constants $r, δ > 0$ and a sequence ${y_{n}} \subset R^{4}$ such that $\begin{matrix} \underset{n \to \infty}{lim inf} \int_{B_{r} (y_{n})} | v_{n} |^{2} ⩾ δ > 0 . \end{matrix}$ Set $u_{n} (\cdot) : = v_{n} (\cdot + y_{n})$ . Then, since $R^{4}$ is invariant by translation, we see that ${u_{n}}$ is a bounded ${(PS)}_{c_{λ}}$ sequence and $\begin{matrix} (3.12) & \underset{n \to \infty}{lim inf} \int_{B_{r} (0)} | u_{n} |^{2} ⩾ δ > 0 . \end{matrix}$ Therefore, Lemma 3.4 implies that there exists $u_{λ} \in H$ such that $u_{n} ⇀ u_{λ}$ in H and $J_{λ}^{'} (u_{λ}) = 0$ . On the other hand, it follows from (3.12) that $\begin{matrix} \underset{n \to \infty}{lim inf} \int_{B_{r} (0)} | u_{λ} |^{2} ⩾ δ > 0, \end{matrix}$ which implies that $u_{λ} \neq 0$ . Hence, $u_{λ} \in N_{λ}$ is a nontrivial critical point of $J_{λ}$ and $c_{λ} ⩽ J_{λ} (u_{λ})$ . It remains to show that $c_{λ} ⩾ J_{λ} (u_{λ})$ . By Lemma 2.2(i), Fatou’s Lemma and the lower semicontinuity of the norm we have $\begin{matrix} c_{λ} & = lim_{n \to \infty} [J_{λ} (u_{n}) - \frac{1}{4} ⟨ J_{λ}^{'} (u_{n}), u_{n} ⟩] \\ = lim_{n \to \infty} [\frac{1}{4} \int_{R^{4}} | \nabla u_{n} |^{2} d x + \frac{1 - θ_{0}}{4} \int_{R^{4}} u_{n}^{2} d x + \int_{R^{4}} (\frac{1}{4} f (u_{n}) u_{n} - F (u_{n}) + \frac{θ_{0}}{4} u_{n}^{2}) d x] \\ ⩾ \underset{n \to \infty}{lim inf} [\frac{1}{4} \int_{R^{4}} | \nabla u_{n} |^{2} d x + \frac{1 - θ_{0}}{4} \int_{R^{4}} u_{n}^{2} d x + \int_{R^{4}} (\frac{1}{4} f (u_{n}) u_{n} - F (u_{n}) + \frac{θ_{0}}{4} u_{n}^{2}) d x] \\ ⩾ \frac{1}{4} \int_{R^{4}} | \nabla u_{λ} |^{2} d x + \frac{1 - θ_{0}}{4} \int_{R^{4}} | u_{λ} |^{2} d x + \int_{R^{4}} (\frac{1}{4} f (u_{λ}) u_{λ} - F (u_{λ}) + \frac{θ_{0}}{4} u_{λ}^{2}) d x \\ = \frac{1}{4} ‖ u_{λ} ‖^{2} + \int_{R^{4}} [\frac{1}{4} f (u_{λ}) u_{λ} - F (u_{λ})] d x \\ = J_{λ} (u_{λ}) - \frac{1}{4} ⟨ J_{λ}^{'} (u_{λ}), u_{λ} ⟩ \\ = J_{λ} (u_{λ}) . \end{matrix}$ Hence, $c_{λ} ⩾ J_{λ} (u_{λ})$ , and so $J_{λ} (u_{λ}) = c_{λ}$ . Thus, $(u_{λ}, ϕ_{u_{λ}})$ is a ground state solution for system ( P λ ). Finally, in order to prove the existence of a positive ground state solution to system ( P λ ), let us consider the following functional $\begin{matrix} J_{λ}^{+} (u) = \frac{1}{2} ‖ u ‖^{2} + \frac{λ}{4} \int_{R^{4}} ϕ_{u} u^{2} d x - \int_{R^{4}} F (u) d x - \frac{1}{4} \int_{R^{4}} {| u^{+} |}^{4} d x, \end{matrix}$ where $u^{+} = max {u, 0}$ and $u^{-} = max {- u, 0}$ . Under the assumption $(f_{0})$ , repeating all the reasonings of this and the preceding section to $J_{λ}^{+}$ , we conclude that there exists $λ^{*} > 0$ such that the system ( P λ ) has a nonnegative ground state solution provided $λ \in (0, λ^{*})$ . Thus, system ( P λ ) has a positive ground state solution due to the strong maximum principle. □

Proof of Theorem 1.3.

Replacing H with $H_{r}$ in the arguments of Sections 2 and 3, by exactly the same process as the proof of Theorem 1.2, we can conclude the existence of a radial positive ground state solution $(u_{λ}, ϕ_{u_{λ}}) \in H_{r} \times D^{1, 2} (R^{4})$ to system ( P λ ).

In the following, we consider the asymptotic behavior of $(u_{λ}, ϕ_{u_{λ}})$ as $λ \to 0$ . Noticing that $λ = 0$ is allowed in the arguments of Section 2. Therefore, there exists $v_{0} \in N_{0}$ such that $\begin{matrix} (3.13) & J_{0}^{'} (v_{0}) = 0 and J_{0} (v_{0}) = c_{0}, \end{matrix}$ where $c_{0} = {inf}_{u \in N_{0}} J_{0} (u)$ , that is, a ground state solution to ( P 0 ).

For any sequence ${λ_{n}} \subset (0, 1]$ with $λ_{n} ↘ 0$ as $n \to 0$ , let $(u_{λ_{n}}, ϕ_{λ_{n}})$ be a radial positive ground state solution of problem ( $P_{λ_{n}}$ ) obtained in Theorem 1.3. By Lemma 3.2, we know that $c_{λ_{n}} < \frac{1}{4} S^{2}$ . Consequently, similar to (3.10) we have $\begin{matrix} \frac{1}{4} S^{2} > c_{λ_{n}} = J_{λ_{n}} (u_{λ_{n}}) - \frac{1}{4} ⟨ J_{λ_{n}}^{'} (u_{λ_{n}}), u_{λ_{n}} ⟩ ⩾ \frac{1 - θ_{0}}{4} ‖ u_{λ_{n}} ‖^{2} . \end{matrix}$ This shows that ${u_{λ_{n}}}$ is bounded in H. Hence, there exists $u_{0} \in H$ such that up to a subsequence $u_{λ_{n}} ⇀ u_{0}$ in H. Since the embedding $H ↪ L^{p} (R^{4})$ is compact for $p \in (2, 4)$ , by using $(f_{1})$ , $(f_{2})$ and Lemma 2.1(3), it is easy to prove that $u_{λ_{n}} \to u_{0}$ strongly in $H_{r}$ as $n \to \infty$ . Furthermore, $\begin{matrix} ⟨ J_{0}^{'} (u_{0}), φ ⟩ = & ⟨ u_{0}, φ ⟩ - \int_{R^{4}} f (u_{0}) φ d x - \int_{R^{4}} | u_{0} |^{2} u_{0} φ d x \\ = & lim_{n \to \infty} [⟨ u_{λ_{n}}, φ ⟩ + λ_{n} \int_{R^{4}} ϕ_{λ_{n}} u_{λ_{n}} φ d x - \int_{R^{4}} f (u_{λ_{n}}) φ d x - \int_{R^{4}} | u_{λ_{n}} |^{2} u_{λ_{n}} φ d x] \\ = & lim_{n \to \infty} ⟨ J_{λ_{n}}^{'} (u_{λ_{n}}), φ ⟩ = 0, \forall φ \in C_{0}^{\infty} (R^{4}) . \end{matrix}$ Thus, $J_{0}^{'} (u_{0}) = 0$ and then $u_{0} \in N_{0}$ and $J_{0} (u_{0}) ⩾ c_{0}$ .

Next, we prove that $J_{0} (u_{0}) = c_{0}$ . By Lemma 2.6, for all $n \in N$ , there exists $t_{n} > 0$ such that $t_{n} v_{0} \in N_{λ_{n}}$ . We claim that ${t_{n}}$ is bounded. Arguing by contradiction suppose that there exists a subsequence of ${t_{n}}$ , still denoted by ${t_{n}}$ such that $t_{n} \to + \infty$ as $n \to \infty$ . Then, since $t_{n} v_{0} \in N_{λ_{n}}$ we have $\begin{matrix} (3.14) & \frac{1}{t_{n}^{2}} ‖ v_{0} ‖^{2} + λ_{n} \int_{R^{4}} ϕ_{v_{0}} v_{0}^{2} d x = \int_{R^{4}} \frac{f (t_{n} v_{0})}{{(t_{n} v_{0})}^{3}} v_{0}^{4} d x + \int_{R^{4}} | v_{0} |^{4} d x . \end{matrix}$ On the other hand, $v_{0} \in N_{0}$ provides $\begin{matrix} (3.15) & ‖ v_{0} ‖^{2} = \int_{R^{4}} f (v_{0}) v_{0} d x + \int_{R^{4}} | v_{0} |^{4} d x . \end{matrix}$ Therefore, for $n \in N$ large enough, combining (3.14) and (3.15) with $(f_{3})$ it holds that $\begin{matrix} (1 - \frac{1}{t_{n}^{2}}) ‖ v_{0} ‖^{2} - λ_{n} \int_{R^{4}} ϕ_{v_{0}} v_{0}^{2} d x & = \int_{R^{4}} [\frac{f (v_{0})}{v_{0}^{3}} - \frac{f (t_{n} v_{0})}{{(t_{n} v_{0})}^{3}}] v_{0}^{4} d x \\ ⩽ θ_{0} (1 - \frac{1}{t_{n}^{2}}) \int_{R^{4}} v_{0}^{2} d x . \end{matrix}$ Passing to the limit in the above inequality, we get $\begin{matrix} ‖ v_{0} ‖^{2} ⩽ θ_{0} \int_{R^{4}} v_{0}^{2} d x < \int_{R^{4}} v_{0}^{2} d x, \end{matrix}$ since $0 < θ_{0} < 1$ . This is a contradiction. Thus we conclude that there exists $M > 0$ such that $\begin{matrix} (3.16) & 0 < t_{n} ⩽ M, \forall n \in N . \end{matrix}$ It follows from Lemma 2.2(2), (2.9), (2.13), (2.11), (3.13) and (3.16) that $\begin{matrix} c_{0} & = J_{0} (v_{0}) \\ = J_{λ_{n}} (v_{0}) - \frac{λ_{n}}{4} \int_{R^{4}} ϕ_{v_{0}} v_{0}^{2} d x \\ ⩾ J_{λ_{n}} (t_{n} v_{0}) + \frac{1 - t_{n}^{4}}{4} ⟨ J_{λ_{n}}^{'} (v_{0}), v_{0} ⟩ + \frac{(1 - θ_{0}) {(1 - t_{n}^{2})}^{2}}{4} ‖ v_{0} ‖^{2} - \frac{λ_{n}}{4} \int_{R^{4}} ϕ_{v_{0}} v_{0}^{2} d x \\ ⩾ c_{λ_{n}} + \frac{1 - t_{n}^{4}}{4} λ_{n} \int_{R^{4}} ϕ_{v_{0}} v_{0}^{2} d x - \frac{λ_{n}}{4} \int_{R^{4}} ϕ_{v_{0}} v_{0}^{2} d x \\ ⩾ c_{λ_{n}} - \frac{t_{n}^{4}}{4} λ_{n} \int_{R^{4}} ϕ_{v_{0}} v_{0}^{2} d x \\ ⩾ c_{λ_{n}} - \frac{M^{4}}{4} λ_{n} \int_{R^{4}} ϕ_{v_{0}} v_{0}^{2} d x, \end{matrix}$ which yields $\begin{matrix} (3.17) & \underset{n \to \infty}{lim sup} c_{λ_{n}} ⩽ c_{0} . \end{matrix}$ Hence, by (2.9) and (3.17) we deduce that $\begin{matrix} c_{0} ⩽ J_{0} (u_{0}) = \underset{n \to \infty}{lim sup} J_{λ_{n}} (u_{λ_{n}}) = \underset{n \to \infty}{lim sup} c_{λ_{n}} ⩽ c_{0} . \end{matrix}$ This shows that $J_{0} (u_{0}) = c_{0}$ . Thus, the proof is completed. □

Footnotes

Acknowledgements

The authors would like to thank the handling editor and the anonymous referee for careful reading the manuscript and suggesting many valuable comments. This work was supported by Natural Science Foundation of China (11671403) and the Mathematics Interdisciplinary Science project of CSU.

References

C.O.

Alves and

L.M.

Barros, Existence and multiplicity of solutions for a class of elliptic problem with critical growth, Monatsh. Math. (2018), to appear. doi:10.1007/s00605-017-1117-z.

Ambrosetti, On Schrödinger–Poisson systems, Milan J. Math. 76(1) (2008), 257–274. doi:10.1007/s00032-008-0094-z.

Azzollini,

d’Avenia and

Pomponio, On the Schrödinger–Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Non Linéair 27(2) (2010), 779–791. doi:10.1016/j.anihpc.2009.11.012.

Bartsch and

Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation, J. Funct. Anal. 117 (1993), 447–460. doi:10.1006/jfan.1993.1133.

Benci and

Donato, An eigenvalue problem for the Schrödinger–Maxwell equations, Topol. Meth. Nonlinear Anal. 11(2) (1998), 283–293, https://projecteuclid.org/euclid.tmna/1476842831 . doi:10.12775/TMNA.1998.019.

Benci and

Donato, Solitary waves of the nonlinear Klein–Gordon equation coupled with the Maxwell equations, Rev. Math. Phys. 14(4) (2002), 409–420. doi:10.1142/S0129055X02001168.

Bieganowski and

Mederski, Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities, Commun. Pure Appl. Anal. 17(1) (2018), 143–161. doi:10.3934/cpaa.2018009.

Brézis and

Nirenberg, Positive solutions of nonlinear elliptic problems involving critical Sobolev exponent, Commun. Pure Appl. Math. 36 (1983), 437–477. doi:10.1002/cpa.3160360405.

Cerami and

Vaira, Positive solutions for some non-autonomous Schrödinger–Poisson systems, J. Differential Equations 248 (2010), 521–543. doi:10.1016/j.jde.2009.06.017.

10.

Chen and

Tian, Infinitely many solutions for Schrödinger–Maxwell equations with indefinite sign subquadratic potentials, Appl. Math. Comput. 226 (2014), 492–502. doi:10.1016/j.amc.2013.10.069.

11.

Chen and

Tang, Ground state sign-changing solutions for a class of Schrödinger–Poisson type problems in

R^{3}

, Z. Angew. Math. Phys. 67(4) (2016), 1–18. doi:10.1007/s00033-016-0695-2.

12.

D’Aprile and

Mugnai, Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134(5) (2004), 893–906. doi:10.1017/S030821050000353X.

13.

G.C.G.

Dos Santos and

G.M.

Figueiredo, Existence of solutions for an NSE with discontinuous nonlinearity, J. Fixed Point Theory Appl. 19(1) (2017), 917–937. doi:10.1007/s11784-017-0404-6.

14.

M.F.

Furtado,

L.A.

Maiay and

E.S.

Medeiros, Positive and nodal solutions for a nonlinear Schrödinger equation with indefinite potential, Adv. Nonlinear Stud. 8(2) (2008), 353–373. doi:10.1515/ans-2008-0207.

15.

Huang,

Rocha and

Chen, Positive and sign-changing solutions of a Schrödinger–Poisson system involving a critical nonlinearity, J. Math. Anal. Appl. 1 (2013), 55–69. doi:10.1016/j.jmaa.2013.05.071.

16.

W.N.

Huang and

X.H.

Tang, Ground-state solutions for asymptotically cubic Schrödinger–Maxwell equations, Mediterr. J. Math. 13 (2016), 3469–3481. doi:10.1007/s00009-016-0697-5.

17.

Khoutir and

Chen, Multiple nontrivial solutions for a nonhomogeneous Schrödinger–Poisson system in

R^{3}

, Electron. J. Qual. Theory Differ. Equ. 28 (2017), 1. doi:10.14232/ejqtde.2017.1.28.

18.

Li,

Li and

Shi, Existence of positive solutions to Schrödinger–Poisson type systems with critical exponent, Commun. Contemp. Math. 16(6) (2014), 1450036. doi:10.1142/S0219199714500369.

19.

Liang,

Xu and

Zhu, Revisit to sign-changing solutions for the nonlinear Schrödinger–Poisson system in

R^{3}

, J. Math. Anal. Appl. 435(1) (2016), 783–799. doi:10.1016/j.jmaa.2015.10.076.

20.

Liu,

Chen and

Wang, Multiplicity for a 4-sublinear Schrödinger–Poisson system with sign-changing potential via Morse theory, Compt. Rendus Math. 354(1) (2016), 75–80. doi:10.1016/j.crma.2015.10.018.

21.

Liu,

Chen and

Yang, Least energy sign-changing solutions for nonlinear Schrödinger equations with indefinite-sign and vanishing potential, Appl. Math. Lett. 53 (2016), 100–106. doi:10.1016/j.aml.2015.10.010.

22.

Liu,

Qi and

Zhao, Existence of a ground state solution for a class of singular elliptic equation without the A-R condition, Nonlinear Anal. Real World Applications 39 (2018), 233–245. doi:10.1016/j.nonrwa.2017.06.012.

23.

Liu,

Guo and

Fang, Multiple semiclassical states for coupled Schrödinger–Poisson equations with critical exponential growth, J. Math. Phys. 56(4) (2015), 1288–1311. doi:10.1063/1.4919543.

24.

Ruiz, The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237(2) (2006), 655–674. doi:10.1016/j.jfa.2006.04.005.

25.

Sánchez and

Soler, Long-time dynamics of the Schrödinger–Poisson–Slater system, J. Stat. Phys. 114 (2004), 179–204. doi:10.1023/B:JOSS.0000003109.97208.53.

26.

Sun,

Chen and

Yang, Positive solutions of asymptotically linear Schrödinger–Poisson systems with a radial potential vanishing at infinity, Nonlinear Anal. Theory, Methods & Applications 74(2) (2011), 413–423. doi:10.1016/j.na.2010.08.052.

27.

Sun and

Ma, Ground state solutions for some Schrödinger–Poisson systems with periodic potentials, J. Differential Equations 260 (2016), 2119–2149. doi:10.1016/j.jde.2015.09.057.

28.

Tang, New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Adv. Nonlinear Stud. 14 (2014), 349–361. doi:10.1515/ans-2014-0208.

29.

Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math. 58(4) (2015), 715–728. doi:10.1007/s11425-014-4957-1.

30.

Tang and

Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations 261 (2016), 2384–2402. doi:10.1016/j.jde.2016.04.032.

31.

Willem, Minimax Theorems, Birkhäuser Boston Inc., Boston, 1996. doi:10.1007/978-1-4612-4146-1.

32.

Xie,

Ma and

Zhang, Bound state solutions of Schrödinger–Poisson system with critical exponent, Discrete Contin. Dyn. Syst. 37(1) (2016), 605–625. doi:10.3934/dcds.2017025.

33.

Xu and

Chen, Multiplicity of small negative-energy solutions for a class of nonlinear Schrödinger–Poisson systems, Appl. Math. Comput. 243 (2014), 817–824. doi:10.1016/j.amc.2014.06.043.

34.

M.H.

Yang and

Z.Q.

Han, Existence and multiplicity results for the nonlinear Schrödinger–Poisson systems, Nonlinear Anal. Real World Applications 13(3) (2012), 1093–1101. doi:10.1016/j.nonrwa.2011.07.008.

35.

Zhang, On ground state and nodal solutions of Schrödinger–Poisson equations with critical growth, J. Math. Anal. Appl. 428(1) (2015), 387–404. doi:10.1016/j.jmaa.2015.03.032.

36.

Zhang,

João and

Squassina, Schrödinger–Poisson systems with a general critical nonlinearity, Commun. Contemp. Math. 19 (2016), 1650028. doi:10.1142/S0219199716500280.