Axial symmetry of solutions to semilinear elliptic equations in unbounded domains

Abstract

In this paper we study the problem of the axial symmetry of solutions of some semilinear elliptic equations in unbounded domains. Assuming that the solutions have Morse index one and that the nonlinearity is strictly convex in the second variable, we are able to prove several symmetry results in Rn and in the exterior of a ball. The case of some bounded domains is also discussed.