Symmetry Results for Reaction-diiusion Equations

This article is concerned with symmetry properties of the solutions of the reaction-diiusion equation u + f (u) = 0 in a bounded connected domain in R N (N = 2; 3; : : :). Of especial interest are nonlinear source terms f of the type f (u) = u p ? u q with 0 q < p 1. Two results are presented. The rst result concerns the solution of a free boundary problem, where the domain is unknown and u and its normal derivative @ n u are required to vanish on the boundary @ of. It is shown that, if f is the sum of a continuous nondecreasing function and a Lipschitz continuous function on 0; 1), then the free boundary problem does not have a positive solution unless is a ball; in this case, any positive solution is radially symmetric around the center of the ball and decreasing with the radial distance from the center. The second result concerns the solution of the Dirichlet problem on a ball in R N , when the nonlinear source term f is continuous, but not necessarily Lipschitz continuous at 0. It is shown that, if f is the sum of a locally Lipschitz continuous function on (0; 1) that is nonincreasing near 0 and a function that is Lipschitz continuous on 0; 1), then any positive solution u is radially symmetric around the center of the ball and decreasing with the radial distance from the center.