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Multiple solutions for a critical fractional elliptic system involving concave–convex nonlinearities

Part of: Integral, integro-differential, and pseudodifferential operators Elliptic equations and systems

Published online by Cambridge University Press:  27 October 2016

Wenjing Chen  and
Shengbing Deng
Wenjing Chen
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, People's Republic of China (wjchen@swu.edu.cn; shbdeng@swu.edu.cn)
Shengbing Deng
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, People's Republic of China (wjchen@swu.edu.cn; shbdeng@swu.edu.cn)
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Extract

We are concerned with the multiplicity of solutions to the system driven by a fractional operator with homogeneous Dirichlet boundary conditions. Namely, using fibering maps and the Nehari manifold, we obtain multiple solutions to the following fractional elliptic system:

where Ω is a smooth bounded set in ℝn, n > 2s, with s ∈ (0, 1); (–Δ)s is the fractional Laplace operator;, λ, μ > 0 are two parameters; the exponent n/(n – 2s) ⩽ q < 2; α > 1, β > 1 satisfy is the fractional critical Sobolev exponent.

Keywords

critical fractional elliptic systemconcave–convex nonlinearitiesfibering mapsNehari manifold

MSC classification

Secondary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations 47G20: Integro-differential operators
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 6 , December 2016 , pp. 1167 - 1193
Copyright
Copyright © Royal Society of Edinburgh 2016 

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