ON THE MULTIPLICITY OF NONNEGATIVE SOLUTIONS WITH A NONTRIVIAL NODAL SET FOR ELLIPTIC EQUATIONS ON SYMMETRIC DOMAINS

摘要

We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. Each nonnegative solution of such a problem is symmetric about H and, if strictly positive, it is also decreasing in the direction orthogonal to H on each side of H. The latter is of course not true if the solution has a nontrivial nodal set. In this paper we prove that for a class of domains, including for example all domains which are convex (in all directions), there can be at most one nonnegative solution with a nontrivial nodal set. For general domains, there are at most finitely many such solutions.