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Positive bound states for fractional Schrödinger-Poisson system with critical exponent

  • *Corresponding author

    *Corresponding author
The work is supported by NSFC grant 11501403, by fund program for the Scientific Activities of Selected Returned Overseas Professionals in Shanxi Province (2018) and by the Natural Science Foundation of Shanxi Province (No. 201901D111085)
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  • In this paper, we consider the following fractional critical Schrödinger-Poisson system without perturbation terms

    $ \begin{equation*} \begin{cases} (-\Delta)^su+V(x)u+K(x)\phi u = |u|^{2_s^{\ast}-2}u & \hbox{in $\mathbb{R}^3$,} \\ (-\Delta)^t\phi = K(x)u^2& \hbox{in $\mathbb{R}^3$,} \end{cases} \end{equation*} $

    where $ s\in(\frac{3}{4},1) $, $ t\in(0,1) $. Under some suitable assumptions on potentials $ V(x) $ and $ K(x) $, we show that the ground state positive solutions do not exist, by using the topological argument, we establish the existence of positive bound state solutions in the range $ (\frac{s}{3}\mathcal{S}_s^{\frac{3}{2s}},\frac{2s}{3}\mathcal{S}_s^{\frac{3}{2s}}) $.

    Mathematics Subject Classification: Primary: 35B38, 35R11; Secondary: 53C35.


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