Existence of positive multi-bump solutions for a Schr\"odinger-Poisson system in $\mathbb{R}^{3}$

TL;DR

This paper proves the existence of positive multi-bump solutions for a Schr"odinger-Poisson system in three-dimensional space, using variational methods under specific potential and nonlinearity conditions.

Contribution

It establishes the existence of positive multi-bump solutions for the Schr"odinger-Poisson system with multiple potential wells and subcritical nonlinearities, which was not previously demonstrated.

Findings

Existence of positive multi-bump solutions in $ ext{R}^3$.

Solutions localized around disjoint potential wells.

Application of variational methods to nonlinear PDEs.

Abstract

In this paper we are going to study a class of Schr\"odinger-Poisson system $$ \left\{ \begin{array}{ll} - \Delta u + (\lambda a(x)+1)u+ \phi u = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3},\\ -\Delta \phi=u^2 \mbox{ in } \,\,\, \mathbb{R}^{3}.\\ \end{array} \right. $$ Assuming that the nonnegative function $a(x)$ has a potential well $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint components $\Omega_1, \Omega_2, ....., \Omega_k$ and the nonlinearity $f(t)$ has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by variational methods.