Symmetry of Nonnegative Solutions of Elliptic Equations via a Result of Serrin

We consider the Dirichlet problem for semilinear elliptic equations on a smooth bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction orthogonal to H. Employing Serrin's result on an overdetermined problem, we show that any nonzero nonnegative solution is necessarily strictly positive. One can thus apply a well-known result of Gidas, Ni and Nirenberg to conclude that the solution is reflectionally symmetric about H and decreasing away from the hyperplane in the orthogonal direction.