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Bound states for fractional Schrödinger-Poisson system with critical exponent

  • * Corresponding author: Shuangjie Peng

    * Corresponding author: Shuangjie Peng
The first author is supported by funding for basic research business in Central Universities (innovative funding projects) (2020CXZZ069). The second author is supported by NSFC grant 1207116
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  • This paper deals with the fractional Schrödinger-Poisson system

    $ \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta )^su+V(x)u+K(x)\phi u = |u|^{2_{s}^{*}-2}u, & \text{in}\ {\Bbb R}^3,\\ (-\Delta)^{t}\phi = K(x)u^2, & \text{in}\ {\Bbb R}^3, \end{array} \right. \end{equation*} $

    where $ s\in (\frac{3}{4}, 1) $, $ t\in(0, 1) $, $ \varepsilon $ is a positive parameter, $ 2_{s}^{*} = \frac{6}{3-2s} $ is the critical Sobolev exponent. $ K(x)\in L^{\frac{6}{2t+4s-3}}({\Bbb R}^3) $, $ V(x)\in L^{\frac{3}{2s}}({\Bbb R}^3) $ and $ V(x) $ is assumed to be zero in some region of $ {\Bbb R}^3 $, which means that the problem is of the critical frequency case. In virtue of a global compactness result in fractional Sobolev space and Lusternik-Schnirelman theory of critical points, we succeed in proving the multiplicity of bound states.

    Mathematics Subject Classification: Primary: 35J47, 35J50.

    Citation:

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