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Multiple positive bound state solutions for a critical Choquard equation

  • * Corresponding author: R. Molle

    * Corresponding author: R. Molle
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  • In this paper we consider the problem

    $ (P_{\lambda})\ \ \ \ \ \ \left\{ \begin{array}{rcl} -\Delta u+V_{\lambda}(x)u = (I_{\mu}*|u|^{2^{*}_{\mu}})|u|^{2^{*}_{\mu}-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N}, \\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array} \right. $

    where $ V_{\lambda} = \lambda+V_{0} $ with $ \lambda \geq 0 $, $ V_0\in L^{N/2}({\mathbb{R}}^N) $, $ I_{\mu} = \frac{1}{|x|^\mu} $ is the Riesz potential with $ 0<\mu<\min\{N, 4\} $ and $ 2^{*}_{\mu} = \frac{2N-\mu}{N-2} $ with $ N\geq 3 $. Under some smallness assumption on $ V_0 $ and $ \lambda $ we prove the existence of two positive solutions of $ (P_\lambda) $. In order to prove the main results, we used variational methods combined with degree theory.

    Mathematics Subject Classification: Primary: 81Q05, 35A15, 35B33.


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