Ground State Sign-Changing Solutions for Critical Schrödinger–Poisson System with Steep Potential Well

Abstract

We investigate a class of nonlinear Schrödinger–Poisson system:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V_\lambda (x)u+ \phi u=|u|^{4}u+|u|^{p-2}u \quad \quad \ &{}\text {in} \ \mathbb {R}^{3},\\ -\Delta \phi =u^2 \quad \quad \quad \quad \ \ \quad \quad \quad \quad \quad \quad &{}\text {in} \ \mathbb {R}^{3}, \end{array}\right. } \end{aligned}$$

where $p\in (5,6)$ and $V_\lambda (x)=\lambda V(x)+1$ with $\lambda >0$. Under some mild assumptions on V, the existence of ground state sign-changing solution is obtained for $\lambda >0$ large enough by adopting constrained minimization arguments and analytical techniques. At the same time, we also study the asymptotic behavior of sign-changing solutions as $\lambda \rightarrow \infty$.