Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition

TL;DR

This paper establishes the existence of periodic solutions for a superlinear fractional elliptic problem without relying on the classical Ambrosetti-Rabinowitz growth condition, using variational methods and a limit process.

Contribution

It introduces a novel approach to find periodic solutions for nonlocal fractional problems under weaker growth conditions than the Ambrosetti-Rabinowitz criterion.

Findings

Existence of nontrivial periodic solutions for the fractional problem.

Extension of solutions via a variational linking theorem.

Limit process as m approaches zero to obtain solutions for the degenerate case.

Abstract

The purpose of this paper is to study $T$-periodic solutions to [(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=f(x,u) &\mbox{in} (0,T)^{N} (P) u(x+Te_{i})=u(x) &\mbox{for all} x \in \R^{N}, i=1, \dots, N where $s\in (0,1)$, $N>2s$, $T>0$, $m> 0$ and $f(x,u)$ is a continuous function, $T$-periodic in $x$ and satisfying a suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition. The nonlocal operator $(-\Delta_{x}+m^{2})^{s}$ can be realized as the Dirichlet to Neumann map for a degenerate elliptic problem posed on the half-cylinder $\mathcal{S}_{T}=(0,T)^{N}\times (0,\infty)$. By using a variant of the Linking Theorem, we show that the extended problem in $\mathcal{S}_{T}$ admits a nontrivial solution $v(x,\xi)$ which is $T$-periodic in $x$. Moreover, by a procedure of limit as $m\rightarrow 0$, we also prove the existence of a nontrivial solution to (P) with $m=0$.