A priori bounds for positive solutions of a semilinear elliptic equation

Chris Cosner; Klaus Schmitt

doi:10.1090/S0002-9939-1985-0796444-0

A priori bounds for positive solutions of a semilinear elliptic equation

Chris Cosner, Klaus Schmitt

Mathematics

Research output: Contribution to journal › Article

9 Scopus citations

Abstract

We consider the semilinear elliptic equation -△u = ƒ(u), x ∈ Ω, subject to zero Dirichlet boundary conditions, where ∈ ⊂ Rⁿ is a bounded domain with smooth boundary and the nonlinearity ƒ assumes both positive and negative values. Under the assumption that ∈ satisfies certain symmetry conditions we establish two results providing lower bounds on the C⁰(∈^̅) norm of positive solutions. The bounds derived are the same one obtains in dimension n = 1.

Original language	English (US)
Pages (from-to)	70-73
Number of pages	4
Journal	Proceedings of the American Mathematical Society
Volume	95
Issue number	1
DOIs	https://doi.org/10.1090/S0002-9939-1985-0796444-0
State	Published - Sep 1985

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

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Cosner, C., & Schmitt, K. (1985). A priori bounds for positive solutions of a semilinear elliptic equation. Proceedings of the American Mathematical Society, 95(1), 70-73. https://doi.org/10.1090/S0002-9939-1985-0796444-0

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AB - We consider the semilinear elliptic equation -△u = ƒ(u), x ∈ Ω, subject to zero Dirichlet boundary conditions, where ∈ ⊂ Rn is a bounded domain with smooth boundary and the nonlinearity ƒ assumes both positive and negative values. Under the assumption that ∈ satisfies certain symmetry conditions we establish two results providing lower bounds on the C0(∈̅) norm of positive solutions. The bounds derived are the same one obtains in dimension n = 1.

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