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

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.