# Ground state solutions of fractional Schr\"odinger equations with potentials and weak monotonicity condition on the nonlinear term

## TL;DR

This paper establishes the existence of ground state solutions for fractional Schrödinger equations under weak monotonicity conditions on the nonlinear term, extending previous results that required stricter assumptions.

## Contribution

It introduces new methods to handle weak monotonicity of the nonlinear term, broadening the class of potentials and nonlinearities for which solutions can be found.

## Key findings

- Existence of ground states for periodic potentials and nonlinearities.
- Infinitely many solutions when the nonlinearity is odd.
- Ground states under coercive or bounded potential well conditions.

## Abstract

In this paper we are concerned with the fractional Schr\"{o}dinger equation $(-\Delta)^{\alpha} u+V(x)u =f(x, u)$, $x\in \rn$, where $f$ is superlinear, subcritical growth and $u\mapsto\frac{f(x, u)}{\vert u\vert}$ is nondecreasing. When $V$ and $f$ are periodic in $x_{1},\ldots, x_{N}$, 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, $u\mapsto\frac{f(x, u)}{\vert u\vert}$ was assumed to be strictly increasing, due to this small change, we are forced to go beyond methods of smooth analysis.