Existence of Solutions for a Schrödinger–Poisson System with Critical Nonlocal Term and General Nonlinearity

The Schrödinger–Poisson system is usually used to describe solitary waves for the nonlinear stationary Schrödinger equations interacting with an electromagnetic field. Since the introduction of the Schrödinger–Poisson system by Benci and Fortunato [1], it has been extensively studied. For more detailed information, we refer the interested readers to [2, 3] and the references therein. In recent years, some researchers extensively studied the Schrödinger–Poisson system with critical growth in an unbounded domain and obtained interesting results under various suitable assumptions (see, e.g., [4–10]). But there are currently only a few results for the following Schrödinger–Poisson systems with critical nonlocal terms in a bounded domain [11–13]:


Introduction and Main Result
The Schrödinger-Poisson system is usually used to describe solitary waves for the nonlinear stationary Schrödinger equations interacting with an electromagnetic field. Since the introduction of the Schrödinger-Poisson system by Benci and Fortunato [1], it has been extensively studied. For more detailed information, we refer the interested readers to [2,3] and the references therein.
(g 3 ) There are constants ρ > 2 and ν > 0 such that ρGðx, tÞ ≤ gðx, tÞt + vt 2 , ∀ðx, tÞ ∈ Ω × ℝ: The main difficulties in the present paper are to estimate the critical value and prove the boundedness of (PS) sequence due to the lack of compactness. In order to overcome the above difficulties, by analytic techniques, we shall give the estimate of critical value of associated functional so that system (2) has at least two nontrivial solutions.
Throughout this paper, we use the following notations: (ii) B r (respectively, ∂B r ) denotes the closed ball (respectively, the sphere) of center zero and radius r, which may vary from line to line (iv) Define the best constant S = inf fkuk 2 : u ∈ H 1 0 ðΩÞ Ð Ω juj 2 * dx = 1g, which is attained by the func-
Then, system (2) possesses at least two distinct nontrivial function pair solutions.
Remark 2. Relative to [11,12], the nonlinearity g is of a pure power form in [11,12], and in the present paper, it is a general nonlinearity. Hence, we make a substantial improvement on the works of the former.

Proof of Main Result
Before proving our Theorem 1, we need the following lemmas. (iv) Hence, according to the standard arguments as those in [1], system (2) can be converted into the following boundary value problem: In order to study the existence of nontrivial solutions to problem (3), we shall firstly consider the existence of nontrivial solutions of the following problem: The energy functional corresponding to (4) is where J is well defined with J ∈ C 1 ðH 1 0 ðΩÞ, ℝÞ and The critical points of the functional J are just weak solutions of problem (4). Let U ε ðxÞ = y ε ðxÞ/C ε define a cutoff function φ ∈ C ∞ 0 ðΩÞ such that where B 2R ð0Þ ⊂ Ω, 0 ≤ φðxÞ ≤ 1 for R < |x | <2R: Put u ε ðxÞ = φðxÞU ε ðxÞ, v ε ðxÞ = u ε ðxÞ/ð Ð Ω ju ε j 2 * dxÞ 1/2 * ; hence, Ð Ω jv ε j 2 * dx = 1.
Proof. First, we prove that fu n g is bounded in H 1 0 ðΩÞ. To prove the boundedness of fu n g, arguing by contradiction, suppose that ku n k ⟶ ∞. Set v n = u n /ku n k. Then, kv n k = 1 and kv n k q ≤ C for 1 ≤ q ≤ 2 * . By (g 3 ), we have where θ = min f2 * , ρg, which implies Passing to a subsequence, we may assume that v n ⟶ v in H 1 0 ðΩÞ, then v n ⟶ v in L q ðΩÞ, 1 ≤ q < 2 * , and v n ⟶ v a.e. in Ω. Hence, it follows from (13) that v ≠ 0 , and where σ > 0 is chosen such that jfx ∈ Ω : jvðxÞj ≥ σgj > 0 and K is sufficiently big constant, which is a contradiction. Thus, fu n g is bounded in H 1 0 ðΩÞ and there exists u ∈ H 1 0 ðΩÞ such that u n ⟶ u, up to a subsequence. Furthermore, J ′ ðuÞ = 0 by the weak continuity of J ′ . If u = 0 in Ω, since the term g ðx, uÞ is subcritical, then hJ ′ðu n Þ, u n i = oð1Þ implies By Lemma 3-(iii), one has It follows from (15) and (16) that If ku n k ⟶ 0, it contradicts c > 0. Therefore, By (15) and (18), we get which contradicts c < 2/ðN + 2ÞS N/2 . Thus, u ≠ 0 and it is a nontrivial solution of problem (4).
Proof. For t ≥ 0, we consider the functions where the above inequality comes from Lemma 3-(iv).