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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A priori bounds for positive solutions of a semilinear elliptic equation

Authors: Chris Cosner and Klaus Schmitt
Journal: Proc. Amer. Math. Soc. 95 (1985), 47-50
MSC: Primary 35J60; Secondary 35B45
MathSciNet review: 796444
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Abstract: We consider the semilinear elliptic equation $ - \Delta u = f(u)$, $ x \in \Omega $, subject to zero Dirichlet boundary conditions, where $ \Omega \subset {{\mathbf{R}}^n}$ is a bounded domain with smooth boundary and the nonlinearity $ f$ assumes both positive and negative values. Under the assumption that $ \Omega $ satisfies certain symmetry conditions we establish two results providing lower bounds on the $ {C^0}(\overline \Omega )$ norm of positive solutions. The bounds derived are the same one obtains in dimension $ n = 1$.

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Article copyright: © Copyright 1985 American Mathematical Society