On multi-bump solutions for a class of Schrödinger-Poisson systems with $ p $-Laplacian in $ \mathbb{R}^{3} $

Abstract

In this article, we consider the following a class of Schrödinger-Poisson systems with $ p $-Laplacian in $ \mathbb{R}^{3} $ of the form: $ \begin{equation*} \left\{ \begin{array}{lll} -\Delta_{p}u+(\lambda b(x)+1)|u|^{p-2}u+\phi|u|^{s-2}u = g(u) &\mbox{in}\ \mathbb{R}^{3}, \\ -\Delta\phi = |u|^{s} &\mbox{in}\ \mathbb{R}^{3}, \end{array} \right. \end{equation*} $ where $ 1 < p < 3 $, $ \frac{p}{2} < s < p $, $ \Delta_{p}u : = div(|\nabla u|^{p-2}\nabla u) $ is the $ p $-Laplacian operator, $ \lambda $ is a positive parameter. Assume that the nonnegative function $ b $ possesses a potential well int$ (b^{-1}(\{0\})) $, which is composed of $ k $ disjoint components $ \Omega_{1}, \Omega_{2}, \cdots, \Omega_{k} $ and consider the nonlinearity $ g $ with subcritical growth. Using the variational methods and Morse iteration technique, the existence of positive multi-bump solutions are obtained.

Keywords

• schr&#246;dinger-poisson system
• $ p $-laplacian
• multi-bump solution
• variational methods