Abstract
We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain $\Omega$. We assume that $\Omega$ 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.
Highlights
In this paper we continue our study of nonnegative solutions of nonlinear elliptic problems of the form
For semilinear equations on smooth domains, this result was proved by Gidas, Ni, and Nirenberg [12]; the extension to the fully nonlinear equations on general symmetric domains is due to Berestycki and Nirenberg [2]
Each nodal domain of u is convex in x1 and symmetric about a hyperplane parallel to H0 (a nodal domain refers to a connected component of {x ∈ Ω : u(x) = 0})
Summary
The uniqueness for the Cauchy problem for elliptic equations implies that any two such solutions coincide on a nonempty open subset. Without any additional conditions on Ω, we can prove that the number of solutions with a nontrivial nodal set is finite and can be estimated above by a quantity derived in an explicit way from geometric properties of Ω.
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