Multi-bump solutions for the magnetic Schrödinger–Poisson system with critical growth

Abstract

In this paper, we are concerned with the following magnetic Schrödinger–Poisson system \begin{align*} \begin{cases} -(\nabla+i A(x))^{2}u+(\lambda V(x)+1)u+\phi u=\alpha f(\left | u\right |^{2})u+\vert u\vert^{4}u,& \text{ in }\mathbb{R}^{3}, \\ -\Delta \phi = u^{2}, & \text{ in } \mathbb{R}^{3}, \end{cases} \end{align*} where $\lambda>0$ is a parameter, $f$ is a subcritical nonlinearity, the potential $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a continuous function verifying some conditions, the magnetic potential $A \in L_{loc}^{2}(\mathbb{R}^{3}, \mathbb{R}^{3}) $. Assuming that the zero set of $V(x)$ has several isolated connected components $\Omega _{1},\dots,\Omega _{k}$ such that the interior of $\Omega _{j}$ is non-empty and $\partial \Omega_{j}$ is smooth, where $j\in \left \{1,\dots,k\right \}$, then for $\lambda>0$ large enough, we use the variational methods to show that the above system has at least $2^{k}-1$ multi-bump solutions.

Keywords

• schrödinger–poisson system
• multi-bump solutions
• magnetic field
• critical growth
• variational methods