The effect of domain shape on the number of positive solutions of certain nonlinear equations (Q1113347)

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.