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

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  • In this paper we are going to study a class of Schrödinger-Poisson system $$ \left\{ \begin{array}{ll} - \Delta u + (\lambda a(x)+1)u+ \phi u = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3},\\ -\Delta \phi=4\pi u^2 \mbox{ in } \,\,\, \mathbb{R}^{3}.\\ \end{array} \right. $$ Assuming that the nonnegative function $a(x)$ has a potential well $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint bounded 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.
    Mathematics Subject Classification: 35J20, 35J65.

    Citation:

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