Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions

Source abstract

This paper, which is the follow-up to part I, concerns the equation ( − Δ ) s v + G ′ ( v ) = 0 (-\Delta )^{s} v+G’(v)=0 in R n \mathbb {R}^{n} , with s ∈ ( 0 , 1 ) s\in (0,1) , where ( − Δ ) s (-\Delta )^{s} stands for the fractional Laplacian—the infinitesimal generator of a Lévy process. When n = 1 n=1 , we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits ± 1 \pm 1 at ± ∞ \pm \infty ) if and only if the potential G G has only two absolute minima in [ − 1 , 1 ] [-1,1] , located at ± 1 \pm 1 and satisfying G ′ ( − 1 ) = G ′ ( 1 ) = 0 G’(-1)=G’(1)=0 . Under the additional hypotheses G ( − 1 ) > 0 G(-1)>0 and G ( 1 ) > 0 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 n\geq 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.