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Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents

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This research was partially supported by the NSFC(11571197, ZR2020MA005)
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  • We consider the following Choquard equation

    $ \label{modelv11} \begin{cases} -(a\!+\!\varepsilon \int_{\Omega}|\nabla u|^2)\Delta u\! = \!\left( \int_{\Omega}\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^\mu}dy\right)|u|^{2^{*}_{\mu}-2}u \!+\! \lambda f(x)|u|^{q-2}u \quad in \quad \Omega,\\ u\! = \!0 \qquad \qquad \qquad \qquad \qquad on \quad \partial\Omega, \end{cases} $

    where $ \lambda $ is a real parameter, $ 2^{*}_{\mu} = \frac{2N-\mu}{N-2}(0<\mu<N) $ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under some suitable assumptions on $ \lambda, \; \mu $, via the constrained minimizer method and concentration compactness principle, we prove that this system has multiple of solutions, and one of which is a positive ground state solution. Moreover, by using an abstract result due to K.-C Chang, we admit infinitely many pairs of distinct solutions. In addition, we prove the nonexistence result by Pohožaev identity when $ \lambda<0 $. The main results extend and complement the earlier works in the literature.

    Mathematics Subject Classification: Primary: 35J05, 35J20, 35J60.

    Citation:

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