关键词: Schrödinger-Poisson systems, localized nodal solutions, perturbation method

Abstract: In this paper, we study the existence of localized nodal solutions for Schrödinger-Poisson systems with critical growth \begin{equation*} \left\{ \begin{aligned} &-\varepsilon^2\Delta v+V(x)v+\lambda \psi v=v^{5}+\mu|v|^{q-2}v, \ \ \ \text{in}\,\,\mathbb{R}^3,\\ &-\varepsilon^2\Delta \psi=v^2, \ \ \ \text{in}\,\,\mathbb{R}^3; \,\,v(x)\rightarrow 0,\,\psi(x)\rightarrow 0\quad\text{as}\,\,|x| \rightarrow\infty. \end{aligned} \right. \end{equation*} We establish, for small $\varepsilon$, the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function via the perturbation method, and employ some new analytical skills to overcome the obstacles caused by the nonlocal term $\varphi_u(x)=\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{u^2(y)}{|x-y|}{\rm d}y$. Our results improve and extend related ones in the literature.

Key words: Schrödinger-Poisson systems, localized nodal solutions, perturbation method