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

Abstract

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: { − ( 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 . $$ \textstyle\begin{cases} -(a+ b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\lambda \phi u= \vert u \vert ^{4}u+ k f(u),&x\in \mathbb{R}^{3}, \\ -\Delta \phi =u^{2},&x\in \mathbb{R}^{3}. \end{cases} $$ 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 ≫ 1 $k\gg1$ , where the nonlinear function f ∈ C ( R , R ) $f\in C(\mathbb{R},\mathbb{R})$ . We also give an analysis on the behavior of u b $u_{b}$ as the parameter b → 0 $b\to 0$ .

Keywords

• Kirchhoff–Schrödinger–Poisson systems
• Nodal solution
• Ground state solution
• Nehari manifold