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 (−Δ)sv + G′(v) = 0 in Rn, with s ∈ (0, 1), where (−Δ)s stands for the fractional Laplacian—the infinitesimal generator of a Lévy process. When n = 1, we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits ±1 at ±∞) 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.