Resonant problems for fractional Laplacian

Yutong Chen; Jiabao Su

doi:10.3934/cpaa.2017008

Article Contents

2017, Volume 16, Issue 1: 163-188. Doi: 10.3934/cpaa.2017008

This issue Previous Article A comparison between random and stochastic modeling for a SIR model Next Article Continuity of cost functional and optimal feedback controls for the stochastic Navier Stokes equation in 2D

Resonant problems for fractional Laplacian

Yutong Chen and
Jiabao Su

School of Mathematical Sciences, Capital Normal University, Beijing 100048, People's Republic of China

Author Bio: E-mail address: chenyutong@cnu.edu.cn; E-mail address: sujb@cnu.edu.cn

Received: January 31, 2016

Revised: July 31, 2016

Published: January 2018

Supported by KZ201510028032 and NSFC11601353,11671026.

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Abstract

In this paper we consider the following fractional Laplacian equation

$ \left\{\begin{array}{ll} (-\Delta).s u=g(x, u) & x\in\Omega,\\ u=0, & x \in \mathbb{R}.N\setminus\Omega,\end{array} \right. $

where $ s\in (0, 1)$ is fixed, $\Omega$ is an open bounded set of $\mathbb{R}.N$, $N > 2s$, with smooth boundary, $(-\Delta).s$ is the fractional Laplace operator. By Morse theory we obtain the existence of nontrivial weak solutions when the problem is resonant at both infinity and zero.

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Mathematics Subject Classification: 49J35, 35A15, 35S15, 45G25, 58E05.

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