NONLINEAR EQUATIONS FOR FRACTIONAL LAPLACIANS II: EXISTENCE, UNIQUENESS, AND QUALITATIVE PROPERTIES OF SOLUTIONS

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.