Positivity and symmetry of nonnegative solutions of semilinear elliptic equations on planar domains

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Abstract

We consider the Dirichlet problem for the semilinear equation δu+f(u)=0 on a bounded domain Ω⊂ℝ N. We assume that Ω is convex in a direction e and symmetric about the hyperplane H={x∈ℝ N:x{dot operator}e=0}. It is known that if N≥2 and Ω is of class C 2, then any nonzero nonnegative solution is necessarily strictly positive and, consequently, it is reflectionally symmetric about H and decreasing in the direction e on the set {x∈Ω:x{dot operator}e>0}. In this paper, we prove the same result for a large class of nonsmooth planar domains. In particular, the result is valid if any of the following additional conditions on Ω holds:(i)Ω is convex (not necessarily symmetric) in the direction perpendicular to e,(ii)Ω is strictly convex in the direction e,(iii)Ω is piecewise-C 1,1.

Original languageEnglish (US)
Pages (from-to)4458-4474
Number of pages17
JournalJournal of Functional Analysis
Volume262
Issue number10
DOIs
StatePublished - May 15 2012

Bibliographical note

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

Keywords

  • Planar domain
  • Positivity
  • Semilinear elliptic equation
  • Symmetry of solutions

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