On critical pseudo-relativistic Hartree equation with potential well

Yu Zheng, Minbo Yang, Zifei Shen

Abstract


The aim of this paper is to investigate the existence and asymptotic behavior of the solutions for the critical pseudo-relativistic Hartree equation $$ \sqrt{-\Delta+m^{2}}u+(\beta V(x)-\lambda)u =\bigg(\int_{\mathbb{R}^{N}}\frac{|u(z)|^{2_{\mu}^{\ast}}}{|x-z|^{\mu}}dz\bigg)|u| ^{2_{\mu}^{\ast}-2}u $$% for $\mathbb{R}^{N}$, where $m, \lambda, \beta\in\mathbb{R}^+$, $0< \mu< N$, $N\geq3$, $2_{\mu}^{\ast}=({2N-\mu})/({N-1})$ plays the role of critical exponent due to the Hardy-Littlewood-Sobolev inequality. By transforming the nonlocal problem into a local one via the Dirichlet-to-Neumann map, we are able to obtain the existence of the solutions by variational methods. Suppose that $0< \lambda< \lambda_{1}(\Omega)$ with $\lambda_{1}(\Omega)$ the first eigenvalue and the parameter $\beta$ is large enough, we can prove the existence of ground state solutions. Furthermore, for any sequences $\beta_{n}\rightarrow\infty$, we can show that the ground state solutions $\{u_{n}\}$ converges to a solution of $$ \sqrt{-\Delta+m^{2}}u-\lambda u= \bigg(\int_{\Omega}\frac{|u(z)|^{2_{\mu}^{\ast}}}{|x-z|^{\mu}}dz\bigg) |u|^{2_{\mu}^{\ast}-2}u \quad \mbox{in } \Omega, $$ where $\Omega :=\mbox{int} V^{-1}(0)$ is a nonempty bounded set with smooth boundary. By the way we also establish the existence and nonexistence results for the ground state solutions of the problems set on bounded domain.

Keywords


Pseudo-relativistic Hartree equation; Brezis-Nirenberg problem; Hardy-Littlewood-Sobolev inequality; critical exponent; potential well

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