Existence and concentration behavior of ground state solutions for magnetic nonlinear Choquard equations

Dengfeng  Lü

doi:10.3934/cpaa.2016014

Article Contents

2016, Volume 15, Issue 5: 1781-1795. Doi: 10.3934/cpaa.2016014

This issue Previous Article A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds Next Article A direct method of moving planes for fractional Laplacian equations in the unit ball

Existence and concentration behavior of ground state solutions for magnetic nonlinear Choquard equations

Dengfeng Lü^1,

1.
School of Mathematics and Statistics, Hubei Engineering University, Xiaogan 432000

Received: September 30, 2015

Revised: December 31, 2015

Published: August 2016

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Abstract

In the present paper, we consider the following magnetic nonlinear Choquard equation \begin{eqnarray} \big(-i\nabla+A(x)\big)^{2}u+\big( g_{0}(x)+\mu g(x)\big)u=\big(|x|^{-\alpha}*|u|^{p}\big)|u|^{p-2}u,\\ u\in H^{1}(\mathbb{R}^{N},\mathbb{C}), \end{eqnarray} where $N\geq 3$, $\alpha\in (0,N)$, $p\in(\frac{2N-\alpha}{N}, \frac{2N-\alpha}{N-2})$, $A(x): {\mathbb{R}}^{N}\rightarrow {\mathbb{R}}^{N}$ is a magnetic vector potential, $\mu>0$ is a parameter, $g_{0}(x)$ and $g(x)$ are real valued electric potential functions on ${\mathbb{R}}^{N}$. Under some suitable conditions, we show that there exists $\mu^{*}>0$ such that the above equation has at least one ground state solution for $\mu\geq\mu^{*}$. Moreover, the concentration behavior of solutions is also studied as $\mu\rightarrow +\infty$.

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Mathematics Subject Classification: Primary: 35J60, 35J10; Secondary: 35Q55.

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