Positivity and symmetry of nonnegative solutions of semilinear elliptic equations on planar domains

摘要

We consider the Dirichlet problem for the semilinear equation δu+f(u)=0 on a bounded domain Ω?? ~N. We assume that Ω is convex in a direction e and symmetric about the hyperplane H={x∈? ~N:x{dot operator}e=0}. It is known that if N≥2 and Ω is of class C ~2, then any nonzero nonnegative solution is necessarily strictly positive and, consequently, it is reflectionally symmetric about H and decreasing in the direction e on the set {x∈Ω:x{dot operator}e>0}. In this paper, we prove the same result for a large class of nonsmooth planar domains. In particular, the result is valid if any of the following additional conditions on Ω holds:(i)Ω is convex (not necessarily symmetric) in the direction perpendicular to e,(ii)Ω is strictly convex in the direction e,(iii)Ω is piecewise-C ~(1,1).