2020 On critical pseudo-relativistic Hartree equation with potential well
Yu Zheng, Minbo Yang, Zifei Shen
Topol. Methods Nonlinear Anal. 55(1): 185-226 (2020). DOI: 10.12775/TMNA.2019.094


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.


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Yu Zheng. Minbo Yang. Zifei Shen. "On critical pseudo-relativistic Hartree equation with potential well." Topol. Methods Nonlinear Anal. 55 (1) 185 - 226, 2020. https://doi.org/10.12775/TMNA.2019.094


Published: 2020
First available in Project Euclid: 6 March 2020

zbMATH: 07199340
MathSciNet: MR4100383
Digital Object Identifier: 10.12775/TMNA.2019.094

Rights: Copyright © 2020 Juliusz P. Schauder Centre for Nonlinear Studies


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Vol.55 • No. 1 • 2020
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