Bound states for fractional Schrödinger-Poisson system with critical exponent
Mengyao Chen, Qi Li, Shuangjie Peng
Abstract
<p style='text-indent:20px;'>This paper deals with the fractional Schrödinger-Poisson system <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ s\in (\frac{3}{4}, 1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ t\in(0, 1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> is a positive parameter, <inline-formula><tex-math id="M4">\begin{document}$ 2_{s}^{*} = \frac{6}{3-2s} $\end{document}</tex-math></inline-formula> is the critical Sobolev exponent. <inline-formula><tex-math id="M5">\begin{document}$ K(x)\in L^{\frac{6}{2t+4s-3}}({\Bbb R}^3) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ V(x)\in L^{\frac{3}{2s}}({\Bbb R}^3) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ V(x) $\end{document}</tex-math></inline-formula> is assumed to be zero in some region of <inline-formula><tex-math id="M8">\begin{document}$ {\Bbb R}^3 $\end{document}</tex-math></inline-formula>, 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.