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Layered solutions for a fractional inhomogeneous Allen-Cahn equation. (English) Zbl 1359.35213

Summary: We consider the problem \[ \varepsilon^{2s} (-\partial_{xx})^s \tilde{u}(\tilde{x}) - V(\tilde{x})\tilde{u}(\tilde{x})(1-\tilde{u}^2(\tilde{x}))=0\quad \text{in}\, \mathbb{R}, \] where \((-\partial_{xx})^s\) denotes the usual fractional Laplace operator, \({\varepsilon > 0}\) is a small parameter and the smooth bounded function \(V\) satisfies \(\mathrm{inf}_{\tilde{x} \in \mathbb{R}}V(\tilde{x}) > 0\). 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 \(V\).

MSC:

35R11 Fractional partial differential equations
35Q35 PDEs in connection with fluid mechanics
35J61 Semilinear elliptic equations
35B45 A priori estimates in context of PDEs
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