Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators

Liang Zhang; X. H.  Tang; Yi Chen

doi:10.3934/cpaa.2017039

Article Contents

2017, Volume 16, Issue 3: 823-842. Doi: 10.3934/cpaa.2017039

This issue Previous Article W^{1, p}-estimates for elliptic equations with lower order terms Next Article On nonexistence of solutions to some nonlinear parabolic inequalities

Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators

1.
School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China
2.
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

^*Corresponding author
^*Corresponding author

Received: April 30, 2016

Revised: December 31, 2016

Published: January 2018

This work is partially supported by the NNSF (No. 11571370,11601508) of China, Natural Science Foundation of Jiangsu Province of China (No. BK20140176), Shandong Provincial Natural Science Foundation, China (No. ZR2014AP011) and the Doctoral Fund of University of Jinan (No.XBS160100118).

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Abstract

In this paper, we consider the following perturbed nonlocal elliptic equation

$\left\{ {\begin{array}{*{20}{l}}{ - {{\cal L}_K}u = \lambda u + f(x,u) + g(x,u),\;\;x \in \Omega ,}\\{u = 0,\;\;x \in \mathbb{R}{^N} \setminus \Omega ,}\end{array}} \right. $

where $\Omega$ is a smooth bounded domain in $\mathbb{R}{^N}$, $\lambda$ is a real parameter and $g$ is a non-odd perturbation term. If $f$ is odd in $u$ and satisfies various superlinear growth conditions at infinity in $u$, infinitely many solutions are obtained in spite of the lack of the symmetry of this problem for any $\lambda\in \mathbb{R}$. The results obtained in this paper may be seen as natural extensions of some classical theorems to the case of nonlocal operators. Moreover, the methods used in this paper can be also applied to obtain some new results for the classical Laplace equation with Dirichlet boundary conditions.

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Mathematics Subject Classification: 35J20, 35J60; Secondary: 47G20.

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