Existence and nonexistence of solutions for generalized quasilinear Kirchhoff–Schrödinger–Poisson system

Abstract

In this paper, we consider the existence and nonexistence of solutions for a class of modified Schrödinger–Poisson system with Kirchhoff-type perturbation by use of variational methods. When nonlinear term $h(u)=|u|^{p-2}u$, $1\leq p<\infty$, the nonexistence of nontrivial solutions of system is demonstrated through Pohožaev identity. When nonlinear term $h(u)$ satisfies appropriate assumptions, taking advantage of critical point theorem, we obtain a positive radial solution and a nontrivial one of system when $g(u)$ satisfies different conditions. Moreover, some convergence properties are established as the parameter $b\rightarrow0$. What is more, the nonexistence of nontrivial solutions in critical case is also proved by use of Pohožaev identity.

Keywords

• variational methods
• pohožaev identity
• critical point theorem
• nonexistence
• radial solution