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

In this paper we are going to study a class of Schrödinger-Poisson system $$ \left\{ \begin{array}{ll} - Δu + (λa(x)+1)u+ ϕu = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3},\\ -Δϕ=u^2 \mbox{ in } \,\,\, \mathbb{R}^{3}.\\ \end{array} \right. $$ Assuming that the nonnegative function $a(x)$ has a potential well $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint components $Ω_1, Ω_2, ....., Ω_k$ and the nonlinearity $f(t)$ has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by variational methods.

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