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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
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\Vert x \right\Vert) = 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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