The effect of the domain shape on the existence of positive solutions of the equation \(\Delta u + u^{2^*-1}= 0\)
The author considers his work on the existence and multiplicity of positive solutions of the Dirichlet problem for \(-\Delta u= u^ p\) in \(\Omega\) where \(p\) is the critical exponent. He obtains here some interesting existence results for certain small perturbations of \(\Omega\) which change the topology of \(\Omega\). He also obtains some multiplicity results. It is interesting that the solutions he obtains are quite different to the ones obtained by domain variation methods [as in the reviewer's paper, \textit{E. N. Dancer}, Bull. Lond. Math. Soc. 20, No. 6, 600-602 (1988; Zbl 0662.35043)], when they can be applied. Sometimes, this enables one to improve the multiplicity results. Reviewer's remark. The non-existence results of Carpio, Comte and Levandonskii mentioned in the introduction becomes less surprising when one realizes that their domains are conformally equivalent to star shaped domains.
scientific article; zbMATH DE number 640268