Sign-changing solutions for the Schrödinger-Poisson system with concave-convex nonlinearities in $ \mathbb{R}^{3} $

Abstract

In this paper, we consider the following Schrödinger-Poisson system $ \begin{equation*} \qquad \left\{ \begin{array}{ll} -\Delta u+V(x)u+\phi u = |u|^{p-2}u+ \lambda K(x)|u|^{q-2}u\ \ \ &\ \rm in\; \mathbb{R}^{3}, \\ -\Delta \phi = u^2 \ \ \ &\ \rm in\; \mathbb{R}^{3}.\ \end{array} \right. \end{equation*} $ Under the weakly coercive assumption on $ V $ and an appropriate condition on $ K $, we investigate the cases when the nonlinearities are of concave-convex type, that is, $ 1 < q < 2 $ and $ 4 < p < 6 $. By constructing a nonempty closed subset of the sign-changing Nehari manifold, we establish the existence of least energy sign-changing solutions provided that $ \lambda\in(-\infty, \lambda_*) $, where $ \lambda_* > 0 $ is a constant.

Keywords

• schrödinger-poisson system
• sign-changing solutions
• concave-convex nonlinearities
• variational method