Symmetry and nonexistence of low Morse index solutions in unbounded domains

Abstract

In this paper we prove symmetry results for classical solutions of semilinear elliptic equations in the whole R N or in the exterior of a ball, N ⩾ 2 , in the case when the nonlinearity is either convex or has a convex first derivative. More precisely we prove that solutions having Morse index j ⩽ N are foliated Schwarz symmetric, i.e. they are axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis. From this we deduce some nonexistence results for positive or sign changing solutions in the case when the nonlinearity does not depend explicitly on the space variable.