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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Instability of nonnegative solutions for a class of semipositone problems


Authors: K. J. Brown and R. Shivaji
Journal: Proc. Amer. Math. Soc. 112 (1991), 121-124
MSC: Primary 35B35; Secondary 35B05, 35J65, 35P30
MathSciNet review: 1043405
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Abstract: We consider the boundary value problem

\begin{displaymath}\begin{gathered}- \Delta u(x) = \lambda f(u(x)),\quad x \in \... ...Bu(x) = 0,\quad x \in \partial \Omega \hfill \\ \end{gathered} \end{displaymath}

where $ \Omega $ is a bounded region in $ {R^N}$ with smooth boundary, $ Bu = \alpha h(x)u + (1 - \alpha )\partial u/\partial n$ where $ \alpha \in [0,1]h:\partial \Omega \to {R^ + }$ with $ h = 1$ when $ \alpha = 1$, $ \lambda > 0,f$ is a smooth function with $ f(0) < 0$ (semipositone), $ f'(u) > 0$ for $ u > 0$ and $ f''(u) \geq 0$ for $ u > 0$. We prove that every nonnegative solution is unstable.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1043405-5
PII: S 0002-9939(1991)1043405-5
Article copyright: © Copyright 1991 American Mathematical Society