Positive solutions of a Kirchhoff–Schrödinger--Newton system with critical nonlocal term

Abstract

This paper deals with the following Kirchhoff–Schrödinger–Newton system with critical growth { − M ( ∫ Ω | ∇ u | 2 d x ) Δ u = ϕ | u | 2 ∗ − 3 u + λ | u | p − 2 u , i n Ω , − Δ ϕ = | u | 2 ∗ − 1 , i n Ω , u = ϕ = 0 , o n ∂ Ω , where Ω ⊂ R N ( N ≥ 3 ) is a smooth bounded domain, M ( t ) = 1 + b t θ − 1 with t > 0 , 1 < θ < N + 2 N − 2 , b > 0 , 1 < p < 2 , λ > 0 is a parameter, 2 ∗ = 2 N 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.