Abstract:

In this paper, we study the following Schr\"odinger-Poisson system

\begin{equation*}

\left\{

\begin{array}{ll}

-\varepsilon^{p}\Delta_{p}u+V(x)|u|^{p-2}u+\phi |u|^{p-2}u=f(u)+|u|^{p^{*}-2}u\quad \mbox{in}\quad \mathbb{R}^{3}, \\

-\varepsilon^{2}\Delta \phi =|u|^{p}\quad\mbox{in}\quad \mathbb{R}^{3},

\end{array}

\right.

\end{equation*}

where $\varepsilon>0$ is a parameter, $\frac{3}{2}<p<3$, $\Delta_{p}u=\text{div}(|\nabla u|^{p-2}\nabla u)$, $p^{*}=\frac{3p}{3-p}$, $V:\mathbb{R}^{3}\rightarrow\mathbb{R}$ is a potential function with a local minimum and $f$ is subcritical growth. Based on the penalization method, Nehari manifold techniques and Ljusternik-Schnirelmann category theory, we obtain the multiplicity and concentration of positive solutions to the above system.

Key words: Schr?dinger-Poisson system\and Positive solution\and Ljusternik-Schnirelmann category theory\and Critical growth\and $p$-Laplacian