Solutions for planar Kirchhoff-Schrödinger-Poisson systems with general nonlinearities

Abstract

Abstract In this paper, we study the following Kirchhoff-type Schrödinger-Poisson systems in R 2 $\mathbb{R}^{2}$ : { − ( a + b ∫ R 2 | ∇ u | 2 d x ) Δ u + V ( x ) u + μ ϕ u = f ( u ) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , $$ \textstyle\begin{cases} - (a+b\int _{{\mathbb{R}}^{2}} \vert \nabla u \vert ^{2}\,\mathrm{d}x ) \Delta u+V(x)u+\mu \phi u=f(u),\quad x\in {\mathbb{R}}^{2}, \\ \Delta \phi =u^{2}, \quad x\in {\mathbb{R}}^{2}, \end{cases} $$ where a , b > 0 $a, b>0$ , V ∈ C ( R 2 , R ) $V\in \mathcal{C}({\mathbb{R}}^{2},{\mathbb{R}})$ and f ∈ C ( R , R ) $f\in \mathcal{C}({\mathbb{R}},{\mathbb{R}})$ . By using variational methods combined with some inequality techniques, we obtain the existence of the least energy solution, the mountain pass solution, and the ground state solutions for the above systems under some general conditions for the nonlinearities. Our results extend and improve the main results in [Chen, Shi, Tang, Discrete Contin. Dyn. Syst. 39 (2019) 5867–5889].

Keywords

• Kirchhoff-type problems
• Schrödinger-Poisson systems
• Variational methods
• Existence of solutions