Layered solutions for a fractional inhomogeneous Allen–Cahn equation

We consider the problem ε 2 s ( - ∂ x x ) s u ~ ( x ~ ) - V ( x ~ ) u ~ ( x ~ ) ( 1 - u ~ 2 ( x ~ ) ) = 0 in R , where ( - ∂ x x ) s denotes the usual fractional Laplace operator, ε > 0 is a small parameter and the smooth bounded function V satisfies inf x ~ ∈ R V ( x ~ ) > 0 . For s ∈ ( 1 2 ,...

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Published inNonlinear differential equations and applications Vol. 23; no. 3
Main Authors Du, Zhuoran, Gui, Changfeng, Sire, Yannick, Wei, Juncheng
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.06.2016
Springer Nature B.V
Springer Verlag
Subjects
Analysis
Analysis of PDEs
Clustering
Mathematics
Mathematics and Statistics
Multilayers
existence
multi-layered solutions
35J61
fractional Allen–Cahn equation
Lyapunov–Schmidt reduction
35B45
fractional Allen Cahn equation
Lyapunov Schntidt reduction
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Summary:We consider the problem ε 2 s ( - ∂ x x ) s u ~ ( x ~ ) - V ( x ~ ) u ~ ( x ~ ) ( 1 - u ~ 2 ( x ~ ) ) = 0 in R , where ( - ∂ x x ) s denotes the usual fractional Laplace operator, ε > 0 is a small parameter and the smooth bounded function V satisfies inf x ~ ∈ R V ( x ~ ) > 0 . For s ∈ ( 1 2 , 1 ) , we prove the existence of separate multi-layered solutions for any small ε , where the layers are located near any non-degenerate local maximal points and non-degenerate local minimal points of function V . We also prove the existence of clustering-layered solutions, and these clustering layers appear within a very small neighborhood of a local maximum point of V .
ISSN:1021-9722
1420-9004
DOI:10.1007/s00030-016-0384-z