On symmetry of nonnegative solutions of elliptic equations

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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 Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about H and decreasing away from the hyperplane in the direction orthogonal to H. For nonnegative solutions, this result is not always true. We show that, nonetheless, the symmetry part of the result remains valid for nonnegative solutions: any nonnegative solution u is symmetric about H. Moreover, we prove that if 0, then the nodal set of u divides the domain Ω into a finite number of reflectionally symmetric subdomains in which u has the usual Gidas-Ni-Nirenberg symmetry and monotonicity properties. We also show several examples of nonnegative solutions with a nonempty interior nodal set.

Original languageEnglish (US)
Pages (from-to)1-19
Number of pages19
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume29
Issue number1
DOIs
StatePublished - 2012

Bibliographical note

Funding Information:
1 Supported in part by NSF grant DMS-0900947.

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