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The effect of domain shape on the number of positive solutions of certain nonlinear equations. (English) Zbl 0662.34025

The author considers how the shape of the bounded domain \(\Omega\) \((\Omega \in R^ m\), \(m>1)\) affects the number of positive solution to the problem (i) \(-\Delta u=\lambda f(u)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), where \(f\in C^ 1[{\mathbb{R}},{\mathbb{R}}]\). For instance, he proves that if \(f(u)=\exp u\), then there are contractible domains \(\Omega\) for which equation (i) has large number of solutions. The case \(f(u)=u^ p\) is also under particular attention. The conditions assuring that the problem (i) (with \(f(u)=u^ p)\) has a unique positive, nondegenerate solution are given. A lot of examples illustrate the obtained results.
Reviewer: D.Bobrowski

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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