Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term

Abstract

In this paper we are concerned with the fractional Schrodinger equation \begin{document}$ (-\Delta)^{\alpha} u+V(x)u = f(x, u) $\end{document} , \begin{document}$ x\in {{\mathbb{R}}^{N}} $\end{document} , where \begin{document}$ f $\end{document} is superlinear, subcritical growth and \begin{document}$ u\mapsto\frac{f(x, u)}{\vert u\vert} $\end{document} is nondecreasing. When \begin{document}$ V $\end{document} and \begin{document}$ f $\end{document} are periodic in \begin{document}$ x_{1},\ldots, x_{N} $\end{document} , we show the existence of ground states and the infinitely many solutions if \begin{document}$ f $\end{document} is odd in \begin{document}$ u $\end{document} . When \begin{document}$ V $\end{document} is coercive or \begin{document}$ V $\end{document} has a bounded potential well and \begin{document}$ f(x, u) = f(u) $\end{document} , the ground states are obtained. When \begin{document}$ V $\end{document} and \begin{document}$ f $\end{document} are asymptotically periodic in \begin{document}$ x $\end{document} , we also obtain the ground states solutions. In the previous research, \begin{document}$ u\mapsto\frac{f(x, u)}{\vert u\vert} $\end{document} was assumed to be strictly increasing, due to this small change, we are forced to go beyond methods of smooth analysis.