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

Vincenzo Ambrosio

doi:10.3934/dcds.2017099

Article Contents

2017, Volume 37, Issue 5: 2265-2284. Doi: 10.3934/dcds.2017099

This issue Previous Article On uniqueness properties of solutions of the Toda and Kac-van Moerbeke hierarchies Next Article Uncountably many planar embeddings of unimodal inverse limit spaces

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

Vincenzo Ambrosio

Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico Ⅱ", via Cinthia 1, 80126 Napoli, Italy

Received: January 31, 2016

Revised: December 31, 2016

Published: May 2017

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Abstract

The purpose of this paper is to study $T$-periodic solutions to

$\left\{ \begin{array}{*{35}{l}} [{{(-{{\Delta }_{x}}+{{m}^{2}})}^{s}}-{{m}^{2s}}]u=f(x,u) & \text{ in }{{(0,T)}^{N}} \\ u(x+T{{e}_{i}})=u(x) & \text{for all }x\text{ }\in {{\mathbb{R}}^{N}},i=1,\ldots ,N \\\end{array} \right. \tag{1}$

where $s∈ (0,1)$, $N> 2s$, $T>0$, $m> 0$ 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 $(-Δ_{x}+m^{2})^{s}$ can be realized as the Dirichlet to Neumann map for a degenerate elliptic problem posed on the half-cylinder $\mathcal{S}_{T}=(0,T)^{N}× (0,∞)$. By using a variant of the Linking Theorem, we show that the extended problem in $\mathcal{S}_{T}$ admits a nontrivial solution $v(x,ξ)$ which is $T$ -periodic in $x$. Moreover, by a procedure of limit as $m\to 0$, we prove the existence of a nontrivial solution to (1) with $m=0$.

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Mathematics Subject Classification: Primary: 49J35, 35A15, 35R11; Secondary: 35J60, 35S05.

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