## Topological Methods in Nonlinear Analysis

### Positive least energy solutions for coupled nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponent

#### Abstract

In this paper, we study the existence and nonexistence of positive least energy solutions of the following coupled nonlinear Schrödinger equations with Choquard type nonlinearities: \begin{equation*} \begin{cases} \displaystyle -\Delta u+\nu_{1}u=\mu_{1}\left(\frac{1}{|x|^{4}}\ast u^{2}\right)u +\beta \left(\frac{1}{|x|^{4}}\ast v^{2}\right) u, & x \in \Omega,\\ -\Delta v+\nu_{2}v=\mu_{2}\left(\frac{1}{|x|^{4}}\ast v^{2}\right)v +\beta\left(\frac{1}{|x|^{4}}\ast u^{2}\right)v, & x \in \Omega,\\ u,v \geq 0 \quad\text{in }\Omega, \qquad u=v=0 \quad \text{on } \partial\Omega. \end{cases} \end{equation*} Here $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain, $-\lambda_{1}(\Omega)< \nu_{1},\nu_{2}< 0, \lambda_{1}(\Omega)$ is the first eigenvalue of $(-\Delta, H_{0}^{1}(\Omega))$, $\mu_{1},\mu_{2}> 0$ and $\beta\neq 0$ is a coupling constant. We show that the critical nonlocal elliptic system has a positive least energy solution under appropriate conditions on parameters via variational methods. For the case in which $\nu_{1}=\nu_{2}$, we obtain the classification of the positive least energy solutions. Moreover, the asymptotic behaviors of the positive least energy solutions as $\beta\rightarrow 0$ are studied.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 53, Number 2 (2019), 623-657.

Dates
First available in Project Euclid: 11 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1557540142

Digital Object Identifier
doi:10.12775/TMNA.2019.014

#### Citation

You, Song; Wang, Qingxun; Zhao, Peihao. Positive least energy solutions for coupled nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponent. Topol. Methods Nonlinear Anal. 53 (2019), no. 2, 623--657. doi:10.12775/TMNA.2019.014. https://projecteuclid.org/euclid.tmna/1557540142