Concentration phenomena for a fractional Choquard equation with magnetic field

Abstract

We consider the following nonlinear fractional Choquard equation $$ \varepsilon^{2s}(-Δ)^{s}_{A/\varepsilon} u + V(x)u = \varepsilon^{μ-N}\left(\frac{1}{|x|^μ}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N}, $$ where $\epsilon >0$ is a parameter, $s\in (0,1)$, $0<\mu <2s$, $N\ge 3$, $(-\Delta {)}_A^s$ is the fractional magnetic Laplacian, $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ is a smooth magnetic potential, $V:{\mathbb{R}}^N\to \mathbb{R}$ is a positive potential with a local minimum and $f$ is a continuous nonlinearity with subcritical growth. By using variational methods we prove the existence and concentration of nontrivial solutions for $\epsilon >0$ small enough.

arXiv admin note: text overlap with arXiv:1801.00199

References in corpus (1)

• Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics