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 Schrödinger equation $(-\Delta {)}^{\alpha }u+V(x)u=f(x,u)$, $x\in \rn$, where $f$ is superlinear, subcritical growth and $u\mapsto \frac{f(x,u)}{|u|}$ is nondecreasing. When $V$ and $f$ are periodic in $x_1,\dots ,x_N$, we show the existence of ground states and the infinitely many solutions if $f$ is odd in $u$. When $V$ is coercive or $V$ has a bounded potential well and $f(x,u)=f(u)$, the ground states are obtained. When $V$ and $f$ are asymptotically periodic in $x$, we also obtain the ground states solutions. In the previous research, $u\mapsto \frac{f(x,u)}{|u|}$ was assumed to be strictly increasing, due to this small change, we are forced to go beyond methods of smooth analysis.