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

Peter Poláčik

doi:10.3934/dcds.2014.34.2657

Article Contents

2014, Volume 34, Issue 6: 2657-2667. Doi: 10.3934/dcds.2014.34.2657

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

Peter Poláčik^1,

1.
School of Mathematics, University of Minnesota, Minneapolis, MN 55455

Received: February 2013

Published: June 2014

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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.

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Mathematics Subject Classification: 35J60, 35B06, 35B05.

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