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# Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

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Volume 18, Issue 1

# Existence and Multiplicity of Solutions for Nonlinear Elliptic Problems with the Fractional Laplacian

Qing-Mei Zhou
/ Ke-Qi Wang
• College of Mechanical and Electrical Engineering Northeast Forestry University Harbin, Heilongjiang, 150040, P. R.CHINA
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Published Online: 2015-02-10 | DOI: https://doi.org/10.1515/fca-2015-0009

## Abstract

In this paper we consider a nonlinear eigenvalue problem driven by the fractional Laplacian. By applying a version of the three-critical-points theorem we obtain the existence of three solutions of the problem in ${H}^{\alpha }\left(\Omega \right)$. In addition the existence of at least two nontrivial solutions are also been obtained when $2<\mu <{2}_{\alpha }^{*}$.

MSC 2010: 35J60; 47J30

Key Words and Phrases: fractional Laplacian; three critical points theorem; fractional Sobolev spaces; Dirichlet problem

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Published Online: 2015-02-10

Citation Information: Fractional Calculus and Applied Analysis, Volume 18, Issue 1, Pages 133–145, ISSN (Online) 1314-2224,

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