Existence and multiplicity of positive solutions for Kirchhoff-Schrödinger-Poisson system with critical growth

Abstract

This paper is concerned with the following Kirchhoff-Schrodinger-Poisson system: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+ b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u + V(x)u+\phi u=\lambda g(x)u^{q-1}+h(x)u^{5}, &{} \ \ \text{ in } \mathbb {R}^{3},\\ -\Delta \phi = u^{2},~u>0,&{} \ \ \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$where $$a>0$$, $$b\ge 0$$, $$q\in [4,6)$$ and $$\lambda >0$$ is a parameter. Under some suitable conditions on V(x), g(x) and h(x), by using the Nehari manifold technique and the Ljusternik-Schnirelmann category theory, we relate the number of positive solutions with the topology of the global maximum set of h when $$\lambda $$ is small enough. Furthermore, with the aid of the Mountain Pass Theorem, we also obtain an existence result for $$\lambda $$ sufficiently large. Recent results from the literature are generally improved and extended.