Multiplicity of solutions for variable-order fractional Kirchhoff equations with nonstandard growth

The aim of this paper is to study a degenerate Kirchhoff-type elliptic problem driven by the fractional Laplace operator with variable order derivative and variable exponents. More precisely, we consider{[u]s(⋅)2(θ−1)(−Δ)s(⋅)u=λa(x)|u|p(x)−2u+b(x)|u|q(x)−2uin Ω,u=0in RN∖Ω, where [u]s(⋅) is the Garli...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2021-09, Vol.501 (1), p.124269, Article 124269
Main Authors: Xiang, Mingqi, Hu, Die, Zhang, Binlin, Wang, Yue
Format: Article
Language:English
Subjects:
Nehari manifold
Nonstandard growth
Variable-order fractional Laplacian
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Summary:The aim of this paper is to study a degenerate Kirchhoff-type elliptic problem driven by the fractional Laplace operator with variable order derivative and variable exponents. More precisely, we consider{[u]s(⋅)2(θ−1)(−Δ)s(⋅)u=λa(x)|u|p(x)−2u+b(x)|u|q(x)−2uin Ω,u=0in RN∖Ω, where [u]s(⋅) is the Garliado seminorm, θ>1, N≥1, s(⋅):RN×RN→(0,1) is a continuous function, λ>0 is a parameter, Ω is a bounded domain in RN with N>2s(x,y) for all (x,y)∈Ω×Ω, (−Δ)s(⋅) is the variable-order fractional Laplacian, a,b∈L∞(Ω) are two positive weight functions, and p,q∈C(Ω‾) with 1
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2020.124269