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Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $

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  • We consider the existence of positive solutions for the following fractional Schrödinger-Poisson system

    $ \begin{equation*} \begin{cases} \varepsilon^{2s}(-\Delta)^{s}u+V(x)u+\phi(x)u = K(x)f(u)+|u|^{2_{s}^{*}-2}u, \ \ & x\in \mathbb{R} ^3, \\ \varepsilon^{2s}(-\Delta)^{s}\phi = u^{2}, \ \ & x \in \mathbb{R} ^3, \end{cases} \end{equation*} $

    where $ s \in (\frac{3}{4}, 1) $, $ \varepsilon $ is a small and positive parameter, $ V $ and $ K $ are nonnegative potential functions. $ 2_{s}^{*} $ is the critical exponent with respect to fractional Sobolev embedding theorem. Under some suitable conditions on the nonlinearity $ f $ and potential functions $ V $ and $ K $, we prove that for $ \varepsilon $ small, the system has a positive ground state solution concentrating around a concrete set related to $ V $ and $ K $. This result generalizes the result for fractional Schrödinger-Poisson system with subcritical exponent by Yu et al. [39] to critical exponent. Moreover, when $ V $ attains its minimum and $ K $ attains its maximum, we also obtain multiple solutions by Ljusternik-Schnirelmann theory.

    Mathematics Subject Classification: Primary: 35J62; Secondary: 35J20, 35J60.


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