Infinitely many sign-changing solutions for the nonlinear Schrödinger–Poisson system

TLDR: In this paper, the existence of sign-changing solutions for the Schrodinger-Poisson system was investigated and invariant sets of descending flow invariants were used to prove that the system has infinitely many sign changing solutions.

Abstract: In this paper, we consider the following Schrodinger–Poisson system $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u+\phi u=f(u)&{}\quad \text{ in }\ \mathbb {R}^3,\\ -\Delta \phi =u^2&{}\quad \text{ in }\ \mathbb {R}^3. \end{array} \right. \end{aligned}$$ We investigate the existence of multiple bound state solutions, in particular sign-changing solutions. By using the method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions. In particular, the nonlinear term includes the power-type nonlinearity $$f(u)=|u|^{p-2}u$$ for the well-studied case $$p\in (4,6)$$ , and the less studied case $$p\in (3,4)$$ , and for the latter case, few existence results are available in the literature.