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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 $\beta\rightarrow0$.

 

How to Cite this Article:
A. Mao, X. Luo, Multiplicity of solutions to linearly coupled Hartree systems with critical exponent, J. Nonlinear Var. Anal. 7 (2023), 173-200.