Existence of Multi-bump Solutions for the Magnetic Schrödinger–Poisson System in $\pmb {{\mathbb {R}}}^{3}$

Abstract

This paper concerns the following magnetic Schrödinger–Poisson system

$$\begin{aligned} {\left\{ \begin{array}{ll} -(\nabla +i A(x))^{2}u+(\lambda V(x)+Z(x))u+\phi u=f(\left| u\right| ^{2})u ,&{} \text { in }{{\mathbb {R}}}^{3}, \\ -\varDelta \phi = u^{2}, &{} \text { in } {\mathbb {R}}^{3}, \end{array}\right. } \end{aligned}$$

where $\lambda$ is a positive parameter, f has subcritical growth, the potentials V,  $Z:{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}$ are continuous functions verifying some conditions, the magnetic potential $A \in L_\mathrm{loc}^{2}({\mathbb {R}}^{3}, {\mathbb {R}}^{3})$. Assuming that the zero set of V(x) has several isolated connected components $\varOmega _{1},\ldots ,\varOmega _{k}$ such that the interior of $\varOmega _{j}$ is non-empty and $\partial \varOmega _{j}$ is smooth, where $j\in \left\{ 1,\ldots ,k\right\}$, then for $\lambda >0$ large enough, we show that the above system has at least $2^{k}-1$ multi-bump solutions by using variational methods.