Electronic Journal of Qualitative Theory of Differential Equations

This paper deals with the following Kirchhoff–Schrödinger–Newton system with critical growth  −M (∫ Ω |∇u|2dx ) ∆u = φ|u|2−3u + λ|u|p−2u, in Ω, −∆φ = |u|2−1, in Ω, u = φ = 0, on ∂Ω, where Ω ⊂ RN(N ≥ 3) is a smooth bounded domain, M(t) = 1 + btθ−1 with t > 0, 1 < θ < N+2 N−2 , b > 0, 1 < p < 2, λ > 0 is a parameter, 2 ∗ = 2N N−2 is the critical Sobolev exponent. By using the variational method and the Brézis–Lieb lemma, the existence and multiplicity of positive solutions are established.

This system is derived from the Schrödinger-Poisson system −∆u + V(x)u + ηϕ f (u) = h(x, u), in R 3 , System as (1.2) has been studied extensively by many researchers because (1.2) has a strong physical meaning, which describes quantum particles interacting with the electromagnetic field generated by the motion. The Schrödinger-Poisson system (also called Schrödinger-Maxwell system) was first introduced by Benci and Fortunato in [6] as a physical model describing a charged wave interacting with its own electrostatic field in quantum mechanic. For more information on the physical aspects about (1.2), we refer the reader to [6,7]. Many recent studies of (1.2) have focused on existence of multiple solutions, ground states, positive and non-radial solutions. When h(x, u) = |u| p−2 u, Alves et al. in [4] considered the existence of ground state solutions for (1.2) with 4 < p < 6. In [10], Cerami and Vaira proved the existence of positive solutions of (1.2) when h(x, u) = a(x)|u| p−2 u with 4 < p < 6 and a(x) is a nonnegative function. The same result was established in [11,18,22,23] for 2 < p < 6. In [20,25,26,28], by using variational methods, the authors proved the existence of ground state solutions of (1.2) with subcritical and critical growths. In addition, the existence of solutions for Schrödinger-Poisson system involving critical nonlocal term has been paid much attention by many authors, we can see [2,13,16,19,24,27] and so on.
In [5], Arora et al. considered a nonlocal Kirchhoff type equation with a critical Sobolev nonlinearity, using suitable variational techniques, the authors showed how to overcome the lack of compactness at critical levels. In [15], by using the variational method and the concentration compactness principle, Lei and Suo established the existence and multiplicity of nontrivial solutions. Luyen and Cuong [21] obtained the existence of multiple solutions for a given boundary value problem, using the minimax method and Rabinowitz's perturbation method. In [29], Zhou, Guo and Zhang combined the variational method and the mountain pass theorem, to get the existence of weak solutions, this time on the Heisenberg group.
Specially, Azzollini, D'Avenia and Vaira [3] studied the following Schrödinger-Newton type system with critical growth in Ω, where Ω ⊂ R N (N ≥ 3) is a smooth bounded domain. By the variational method, they obtained the existence and nonexistence results of positive solutions when N = 3 and the existence of solutions in both the resonance and the non-resonance case for higher dimensions. Lei and Gao [14] considered the Schrödinger-Newton system with sign-changing potential in Ω, where Ω ⊂ R 3 is a smooth bounded domain, 1 < p < 2, f λ = λ f + + f − , λ > 0, f ± = max{± f , 0}. By using the variational method and analytic techniques, the authors proved the existence and multiplicity of positive solutions.
In [17], Li et al. proved the existence, nonexistence and multiplicity of positive radially symmetric solutions for the following Schrödinger-Poisson system where p ∈ (2, 6), λ ∈ R and µ ≥ 0 are parameters. With the help of the Lax-Milgram theorem, for every u ∈ H 1 0 (Ω), the second equation of system (1.1) has a unique solution ϕ u ∈ H 1 0 (Ω), we substitute ϕ u to the first equation of system (1.1), then system (1.1) transforms into the following equation (1. 3) The variational functional associated with (1.3) is defined by We say that u ∈ H 1 0 (Ω) is a weak solution of (1.3), for all ψ ∈ H 1 0 (Ω), then u satisfies Our technique based on the Ekeland variational principle and the mountain pass theorem. Since system (1.1) contains a nonlocal critical growth term, which leads to the cause of the lack of compactness of the embedding H 1 0 (Ω) → L 2 * (Ω) and the Palais-Smale condition for the corresponding energy functional could not be checked directly. Then we overcome the compactness by using the Brézis-Lieb lemma. Now we state our main result.
Throughout this paper, we make use of the following notations: • C, C 1 , C 2 , . . . denote various positive constants, which may vary from line to line.
• We denote by S ρ (respectively, B ρ ) the sphere (respectively, the closed ball) of center zero and radius ρ, • Let S be the best constant for Sobolev embedding H 1 0 (Ω) → L 2 * (Ω), namely
We say that I λ satisfies (PS) c condition if every (PS) c sequence of I λ has a convergent subsequence in H 1 0 (Ω). Lemma 2.4. Assume that 1 < θ < N+2 N−2 and 1 < p < 2, the functional I λ satisfies the (PS) c condition Proof. Let {u n } ⊂ H 1 0 (Ω) be a (PS) sequence for I λ at the level c, that is Combining with (2.1) and (2.2), we have Therefore {u n } is bounded in H 1 0 (Ω) for all 1 < p < 2. Thus, we may assume up to a subsequence, still denoted by {u n }, that there exists u ∈ H 1 0 (Ω) such that as n → ∞. By (2.1) and the Young inequality, one has Letting η = p(2 * −2) 2(2 * −1)−p and D = Next, we prove that u n → u strongly in H 1 0 (Ω). Set w n = u n − u and lim n→∞ ∥w n ∥ = l, by using the Brézis-Lieb lemma [9], we have
Applying the Sobolev inequality, we get Thus, by (2.8), we can deduce that which implies that l ≥ S N 4 as n → ∞. Since I(u n ) = c + o(1), we obtain Hence, there holds as n → ∞. This is a contradiction. Hence, we can conclude that u n → u in H 1 0 (Ω). The proof is complete.