Anmin Mao, Xiaorong Luo, Multiplicity of solutions to linearly coupled Hartree systems with critical exponent

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DOI: 10.23952/jnva.7.2023.2.01

Volume 7, Issue 2, 1 April 2023, Pages 173-200

Abstract. We consider the existence multiple solutions to the linearly
coupled elliptic system
$-\Delta u+\lambda_{1}u=f(x)\left(\displaystyle\int_{\Omega}\frac{|u(y)|^{2^*_\mu}}{|x-y|^\mu}dy\right)|u|^{2^*_\mu-2}u + \beta v,$ in $\Omega,$
$-\Delta v+\lambda_{2}v=g(x)\left(\displaystyle\int_{\Omega}\frac{|v(y)|^{2^*_\mu}}{|x-y|^\mu}dy\right)|v|^{2^*_\mu-2}v + \beta u,$ in $\Omega,$
$u,v\geq0,$ in $\Omega,$
$u,v=0,$ on $\partial\Omega,$
where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^{N} ~(N\geq3)$, $0\textless\mu\textless \min\{4,N\}$, $\lambda_1,\lambda_2>-\lambda_1(\Omega)$ are constants, $\lambda_{1}(\Omega)$ is the first eigenvalue of $(-\Delta, H^1_0(\Omega))$, $\beta\in\mathbb{R}$ is a coupling parameter, $f,~g\in L^{\infty}(\Omega)$ are nonnegative, and $2^{*}_{\mu}=\frac{2N-\mu}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We prove that the system has a positive ground state solution by mountain pass theorem for small $\beta>0$. By a perturbation argument, when $\lambda_{1}, \lambda_2\in(-\lambda_{1}(\Omega),0)$, comparing with the mountain pass type solution, another positive higher energy solution is obtained when $|\beta|$ is small. In addition, the asymptotic behaviours of these solutions are analyzed as $\beta\rightarrow0$.