On Concentration of least energy solutions for magnetic critical Choquard equations

Abstract

In the present paper, we consider the following magnetic nonlinear Choquard equation $$\left\{\begin{array}{llc}& (-i\nabla +A(x){)}^{2}u+\mu g(x)u=\lambda u+(|x{|}^{-\alpha }\ast |u{|}^{2_{\alpha }^{\ast }})|u{|}^{2_{\alpha }^{\ast }-2}u,u>0\text{in}{\mathbb{R}}^n,& u\in H^1({\mathbb{R}}^n,\mathbb{C})\end{array}\right\}.$$ where $n\ge 4$, $2_{\alpha }^{\ast }=\frac{2n-\alpha }{n-2}$, $\lambda >0$, $\mu \in \mathbb{R}$ is a parameter, $\alpha \in (0,n)$, $A(x):{\mathbb{R}}^n\to {\mathbb{R}}^n$ 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 $\mu \to +\infty$.

17 pages

References in corpus (2)

• Nodal solutions for the Choquard equation
• On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation