On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains

We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain ω We assume that 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 nitely many such solutions.