Capítols de llibrehttp://hdl.handle.net/2117/798182024-03-29T09:22:36Z2024-03-29T09:22:36ZNonlinear equations for fractional laplacians II: existence, uniqueness, and qualitative properties of solutionsCabré Vilagut, XavierSire, Yannickhttp://hdl.handle.net/2117/767872021-03-29T07:49:28Z2015-09-15T10:25:04ZNonlinear equations for fractional laplacians II: existence, uniqueness, and qualitative properties of solutions
Cabré Vilagut, Xavier; Sire, Yannick
This paper, which is the follow-up to part I, concerns the equation (-Delta)(s)v + G'(v) = 0 in R-n, with s is an element of (0, 1), where (-Delta)(s) stands for the fractional Laplacian-the infinitesimal generator of a Levy process.; When n = 1, we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits +/- 1 at +/-infinity) if and only if the potential G has only two absolute minima in [-1, 1], located at +/- 1 and satisfying G'(-1) = G'(1) = 0. Under the additional hypotheses G ''(-1) > 0 and G ''(1) > 0, we also establish its uniqueness and asymptotic behavior at infinity. Furthermore, we provide with a concrete, almost explicit, example of layer solution.; For n >= 1, we prove some results related to the one-dimensional symmetry of certain solutions-in the spirit of a well-known conjecture of De Giorgi for the standard Laplacian.
2015-09-15T10:25:04ZCabré Vilagut, XavierSire, YannickThis paper, which is the follow-up to part I, concerns the equation (-Delta)(s)v + G'(v) = 0 in R-n, with s is an element of (0, 1), where (-Delta)(s) stands for the fractional Laplacian-the infinitesimal generator of a Levy process.; When n = 1, we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits +/- 1 at +/-infinity) if and only if the potential G has only two absolute minima in [-1, 1], located at +/- 1 and satisfying G'(-1) = G'(1) = 0. Under the additional hypotheses G ''(-1) > 0 and G ''(1) > 0, we also establish its uniqueness and asymptotic behavior at infinity. Furthermore, we provide with a concrete, almost explicit, example of layer solution.; For n >= 1, we prove some results related to the one-dimensional symmetry of certain solutions-in the spirit of a well-known conjecture of De Giorgi for the standard Laplacian.