On Concentration of least energy solutions for magnetic critical Choquard equations

TL;DR

This paper investigates the existence and concentration behavior of least energy solutions for a magnetic critical Choquard equation with nonlinear nonlocal terms, using variational methods in high-dimensional space.

Contribution

It establishes the existence of least energy solutions and analyzes their concentration behavior as the parameter 0 increases, which is a novel contribution for this class of equations.

Findings

Existence of least energy solutions under certain conditions.

Solutions concentrate as 0 + in the domain.

Provides insights into the effect of magnetic potential and parameters.

Abstract

In the present paper, we consider the following magnetic nonlinear Choquard equation $$ \left\{ \begin{array}{ll} & (-i \nabla+A(x))^2u + \mu g(x)u = \lambda u + (|x|^{-\alpha} * |u|^{2^*_\alpha})|u|^{2^*_\alpha-2}u ,\; u>0 \;\text{in} \; \mathbb{ R}^n , & u \in H^1(\mathbb{R}^n, \mathbb{ C}) \end{array} \right\}.$$ where $n \geq 4$, $2^*_\alpha= \frac{2n-\alpha}{n-2}$, $\lambda>0$, $\mu \in \mathbb{ R}$ is a parameter, $\alpha \in (0,n)$, $A(x): \mathbb{R}^n \rightarrow \mathbb{ R}^n$ is a magnetic vector potential and $g(x)$ is a real valued potential function on $\mathbb{R}^n$. Using variational methods, we establish the existence of least energy solution under some suitable conditions. Moreover, the concentration behavior of solutions is also studied as $\mu \rightarrow +\infty$.