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

This paper concerns the following magnetic Schrodinger–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.