On Concentration of least energy solutions for magnetic critical Choquard equations

Abstract

In the present paper, we consider the following magnetic nonlinear Choquard equation $$ \left\{ \begin{array}{ll} & (-i \nabla+A(x))^2u + μg(x)u = λu + (|x|^{-α} * |u|^{2^*_α})|u|^{2^*_α-2}u ,\; u>0 \;\text{in} \; \mathbb{ R}^n , & u \in H^1(\mathbb{R}^n, \mathbb{ C}) \end{array} \right\}.$$ where $n \geq 4$, $2^*_α= \frac{2n-α}{n-2}$, $λ>0$, $μ\in \mathbb{ R}$ is a parameter, $α\in (0,n)$, $A(x): \mathbb{R}^n \rightarrow \mathbb{ R}^n$ 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 $μ\rightarrow +\infty$.