Symmetry Results for Solutions of Semilinear Elliptic Equations with Convex Nonlinearities

Abstract

In this paper, we study the symmetry properties of the solutions of the semilinear elliptic problem {−Δu=f(x,u)in Ωu=g(x)on ∂Ω, where Ω is a bounded symmetric domain in RN, N⩾2, and f:Ω×R→R is a continuous function of class C1 in the second variable, g is continuous and f and g are somehow symmetric in x. Our main result is to show that all solutions of the above problem of index one are axially symmetric when Ω is an annulus or a ball, g≡0 and f is strictly convex in the second variable. To do this, we prove that the nonnegativity of the first eigenvalue of the linearized operator in the caps determined by the symmetry of Ω is a sufficient condition for the symmetry of the solution, when f is a convex function.