Abstract
We consider the problem
where \({(-\partial_{xx})^s}\) denotes the usual fractional Laplace operator, \({\varepsilon > 0}\) is a small parameter and the smooth bounded function V satisfies \({{\rm inf}_{\tilde{x} \in \mathbb{R}}V(\tilde{x}) > 0}\). For \({s\in(\frac{1}{2},1)}\), we prove the existence of separate multi-layered solutions for any small \({\varepsilon}\), 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.
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Du, Z., Gui, C., Sire, Y. et al. Layered solutions for a fractional inhomogeneous Allen–Cahn equation. Nonlinear Differ. Equ. Appl. 23, 29 (2016). https://doi.org/10.1007/s00030-016-0384-z
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DOI: https://doi.org/10.1007/s00030-016-0384-z