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

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Abstract

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.

Original languageEnglish (US)
Pages (from-to)2657-2667
Number of pages11
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume34
Issue number6
DOIs
StatePublished - Jun 2014

Keywords

  • Multiplicity
  • Nodal set
  • Nonlinear elliptic equations
  • Symmetry

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