Concentration results for a magnetic Schrödinger-Poisson system with critical growth

Abstract

This paper is concerned with the following nonlinear magnetic Schrödinger-Poisson type equation $$\begin{array}{} \displaystyle \left\{ \begin{aligned} &\Big(\frac{\epsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u=f(|u|^{2})u+\vert u\vert^{4}u \quad \hbox{in }\mathbb{R}^3,\\ &u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}), \end{aligned} \right. \end{array}$$ where ϵ > 0, V : ℝ3 → ℝ and A : ℝ3 → ℝ3 are continuous potentials, f : ℝ → ℝ is a subcritical nonlinear term and is only continuous. Under a local assumption on the potential V, we use variational methods, penalization technique and Ljusternick-Schnirelmann theory to prove multiplicity and concentration of nontrivial solutions for ϵ > 0 small.

Degenerate Fractional Kirchhoff-Type System with Magnetic Fields and Upper Critical Growth

“…Ji and Rȃdulescu [13] obtained the multiplicity and concentration properties of solutions for a class of nonlinear magnetic Schrödinger equation by using variational methods, penalization techniques, and the Ljusternik-Schnirelmann theory. For more interesting results, we refer to [14,28,44,48]. Recently, many researchers have paid attention to the equations with fractional magnetic operator.…”

On the ground states for the X‐ray free electron lasers Schrödinger equation

“…$$ where $$$ \mu \in \mathbb{R} $$$ and $$$ K(x) $$$ is a real function. For the studies on the magnetic Schrödinger‐Poisson system, we refer to Ambrosio, 13 Liu and Ji, 14 and Zhu and Sun 15 and the references therein. For the magnetic Schrödinger‐Poisson system with Coulomb force, it seems that there is no result for the existence of the solution as far as we know.…”