Ground state and nodal solutions for critical Kirchhoff–Schrödinger–Poisson systems with an asymptotically 3-linear growth nonlinearity

Abstract

In this paper, we consider the existence of a least energy nodal solution and a ground state solution, energy doubling property and asymptotic behavior of solutions of the following critical problem: $$ \textstyle\begin{cases} -(a+ b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\lambda \phi u= \vert u \vert ^{4}u+ k f(u),&x\in \mathbb{R}^{3}, \\ -\Delta \phi =u^{2},&x\in \mathbb{R}^{3}. \end{cases} $$ { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u + λ ϕ u = | u | 4 u + k f ( u ) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 . By nodal Nehari manifold method, for each $b>0$ b > 0 , we obtain a least energy nodal solution $u_{b}$ u b and a ground-state solution $v_{b}$ v b to this problem when $k\gg1$ k ≫ 1 , where the nonlinear function $f\in C(\mathbb{R},\mathbb{R})$ f ∈ C ( R , R ) . We also give an analysis on the behavior of $u_{b}$ u b as the parameter $b\to 0$ b → 0 .