The Benci–Cerami problem for the fractional Choquard equation with critical exponent

In this paper we study the existence of bound states for the fractional Choquard equation (-Δ)su+V(x)u=(Iα∗|u|2α,s∗)|u|2α,s∗-2u,x∈RN,u∈Ds,2(RN),u(x)>0,x∈RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^su+V(x)u=(I_{\alpha }*|u|^{2^*_{\alpha ,s}})|u|^{2^*_{\alpha ,s}-2}u,&{}x\in {\mathbb {R}}^N, \\ u\in D^{s,2}({\mathbb {R}}^N),~~u(x)>0,&{}x\in {\mathbb {R}}^N, \end{array}\right. \end{aligned}$$\end{document}where Iα(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{\alpha }(x)$$\end{document} is the Riesz potential, s∈(0,1),N>2s,0<α<min{N,4s},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in (0,1), N>2s, 0<\alpha <\min \{N,4s\},$$\end{document} and 2α,s∗=2N-αN-2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^*_{\alpha ,s}=\frac{2N-\alpha }{N-2s}$$\end{document} is the fractional Hardy–Littlewood–Sobolev critical exponent, (-Δ)s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^s$$\end{document} is the fractional Laplacian operator. We prove some new nonlocal versions of concentration-compactness and global compactness results. Under some suitable assumptions on the potential function V introduced by Benci and Cerami, we show the existence of high energy solution. This study can be considered as a counterpart of the Benci–Cerami problem in the context of high energy solutions for the fractional Choquard equations in the whole space RN.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^N.$$\end{document}