Positive Solutions with High Energy for Fractional Schrödinger Equations

In this paper, we study the Schrödinger equations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(- \Delta)^s}u + V(x)u = a(x)|u{|^{p - 2}}u + b(x)|u{|^{q - 2}}u,\,\,\,\,\,x \in {\mathbb{R}^N},$$\end{document} where 0 < s < 1, 2 < q < p < 2s*,2s* is the fractional Sobolev critical exponent. Under suitable assumptions on V, a and b for which there may be no ground state solution, the existence of positive solutions are obtained via variational methods.