Symmetry results for reaction-diffusion equations
Hans G. Kaper, Man Kam Kwong, Yi Li · Differential and Integral Equations · 1993
This article is concerned with symmetry properties of the solutions of the reaction-diffusion equation ~u + f(u) = 0 in a bounded connected domain n in JRN (N = 2, 3, ... ).Of especial interest are nonlinear source terms f of the type f(u) =uP-uq with 0 :::; q < p :::; 1.Two results are presented.The first result concerns the solution of a free boundary problem, where the domain n is unknown and u and its normal derivative 8nu are required to vanish on the boundary 80 of n.It is shown that, if f is the sum of a continuous nondecreasing function and a Lipschitz continuous function on [0, oo), then the free boundary problem does not have a positive solution unless n 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 IRN, when the nonlinear source term f is continuous, but not necessarily Lipschitz continuous at 0. It is shown that, iff is the sum of a locally Lipschitz continuous function on (O,oo) that is nonincreasing near 0 and a function that is Lipschitz continuous on [0, oo), then any positive solution u is radially symmetric around the center of the ball and decreasing with the radial distance from the center.