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

Abstract

<abstract><p>In this article, we consider the following a class of Schrödinger-Poisson systems with $ p $-Laplacian in $ \mathbb{R}^{3} $ of the form:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{lll} -\Delta_{p}u+(\lambda b(x)+1)|u|^{p-2}u+\phi|u|^{s-2}u = g(u) &amp;\mbox{in}\ \mathbb{R}^{3}, \\ -\Delta\phi = |u|^{s} &amp;\mbox{in}\ \mathbb{R}^{3}, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ 1 &lt; p &lt; 3 $, $ \frac{p}{2} &lt; s &lt; 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.</p></abstract>