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Proceedings of the American Mathematical Society

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

 
 

 

Monotonicity of the forcing term and existence of positive solutions for a class of semilinear elliptic problems


Author: Gadam Sudhasree
Journal: Proc. Amer. Math. Soc. 113 (1991), 415-418
MSC: Primary 35B05; Secondary 35J65
DOI: https://doi.org/10.1090/S0002-9939-1991-1059637-6
MathSciNet review: 1059637
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Abstract: We study the existence of positive solutions to the equation $\Delta u + f(u) + \lambda g(\left \| x \right \|) = 0$ in the unit ball in ${\mathbb {R}^N}$ with Dirichlet boundary conditions, where $f$ is superlinear with $f(0) = 0$ and $\lambda$ is a real parameter. We prove that if $g$ is monotonically increasing, then there exists an $\alpha < 0$ such that for $\lambda < \alpha$ the above equation has no positive solution. This is in contrast to the case of $g$ monotonically decreasing, where positive solutions exist for all negative values of $\lambda$.


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Article copyright: © Copyright 1991 American Mathematical Society