Concentration phenomena for a fractional Choquard equation with magnetic field

We consider the following nonlinear fractional Choquard equation\[\epsilon^{2s} (-\Delta)^s_{A / \epsilon} u+V(x)u = \epsilon^{\mu-N}\left(\frac{1}{{\lvert x \rvert}^{\mu}} \ast F({\lvert u \rvert}^2) \right)f({\lvert u \rvert}^2) u \: \textrm{in} \: \mathbb{R}^N,\]where $\epsilon \gt 0$ is a parameter, $s \in (0, 1) , 0 \lt \mu \lt 2s , N \geq 3 , (-\Delta)^s_A$ 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 $\epsilon \gt 0$ small enough.

Keywords

fractional Choquard equation, fractional magnetic Laplacian, penalization method

2010 Mathematics Subject Classification

Primary 35A15, 35R11. Secondary 45G05.

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Received 4 June 2018

Published 14 March 2019