Symmetry results for solutions of equations involving zero-order operators

Abstract

In this note, we study symmetry results of solutions to equation (E) -I-epsilon[u] = f (u) in B-1 with the condition u = 0 in (B) over bar (c)(1), where I-epsilon[u](x) = integral(RN) u(y)-u (x)/epsilon(N+2 sigma) +vertical bar y-x vertical bar(N+2 sigma) dy, with epsilon > 0 and sigma is an element of (0, 1), is a zero -order nonlocal operator, which approaches the fractional Laplacian when epsilon -> 0. The function f is locally Lipschitz continuous. We analyzed that the symmetry properties of solutions depend on the Lipschitz constant of f. When the Lipschitz constant is controlled by C-N,sigma epsilon(-2 sigma), any solution u is an element of C((B) over bar (1)) of (E) satisfying u > c in B-1 and u =c on partial derivative B-1 is radially symmetric. (c) 2015 Academie des sciences.