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Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents

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The third author is supported by NSFC grant 11671364 and the Fourth author is supported by NSFC grant 11571317

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  • In this paper we are interested in the following critical coupled Hartree system

    $ \left\{\begin{array}{l} (-\Delta)^{s} \tilde{u}+\lambda_{1}\tilde{u} = \alpha_{1}\int_{\Omega}\frac{|\tilde{u}(z)|^{2_{\mu}^{\ast}} }{|x-z|^{\mu}}dz|\tilde{u}|^{2_{\mu}^{\ast}-2}\tilde{u}\\ \quad +\beta\int_{\Omega}\frac{|\tilde{v}(z)|^{2_{\mu}^{\ast}}} {|x-z|^{\mu}}dz|\tilde{u}|^{2_{\mu}^{\ast}-2}\tilde{u}, \ \ &&\mbox{in} \ \Omega,\\ (-\Delta)^{s} \tilde{v}+\lambda_{2}\tilde{v} = \alpha_{2}\int_{\Omega}\frac{|\tilde{v}(z)|^{2_{\mu}^{\ast}} }{|x-z|^{\mu}}dz|\tilde{v}|^{2_{\mu}^{\ast}-2}\tilde{v}\\ \quad +\beta\int_{\Omega}\frac{|\tilde{u}(z)|^{2_{\mu}^{\ast}}} {|x-z|^{\mu}}dz|\tilde{v}|^{2_{\mu}^{\ast}-2}\tilde{v}, \ \ &&\mbox{in} \ \Omega,\\ \tilde{u} = \tilde{v} = 0, \ \ && \mbox{on} \ \partial \Omega, \end{array} \right. $

    where $ 0<s<1 $, $ \alpha_{1}, \alpha_{2}>0 $, $ \beta\neq0 $, $ 4s<\mu<N $, $ 2_{\mu}^{\ast} = (2N-\mu)/(N-2s) $, $ \Omega\subset\mathbb{R}^N(N\geq3) $ is a smooth bounded domain, $ -\lambda_{1}(\Omega)<\lambda_{1}, \lambda_{2}<0 $ with $ \lambda_{1}(\Omega) $ the first eigenvalue of $ (-\Delta)^{s} $ under the Dirichlet boundary condition. Assume that the nonlinearity and the coupling terms are both of the upper critical growth due to the Hardy–Littlewood–Sobolev inequality, by applying the Dirichlet-to-Neumann map, we are able to obtain the existence of the ground state solution of the critical coupled Hartree system.

    Mathematics Subject Classification: 35J25, 35J60, 35A15.


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