Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains
The paper is concerned with the following problem: \[ (*)\quad \Delta u+u^{2^*-1}=0\text{ in } \Omega;\quad u>0\text{ in } \Omega;\quad u=0\text{ on } \partial \Omega, \] where \(\Omega\) is a bounded domain in \(R^ n\) with \(n\geq 3\) and \(2^*=2n/(n-2)\) is the critical Sobolev exponent. A well known result of Pohozaev states that the problem has no solution if the domain \(\Omega\) is star-shaped; on the other hand, Bahri and Coron have proved that (*) has at least one solution if \(\Omega\) has ``non trivial'' topology. In this paper it is proved that for every positive integer k there exists a contractible bounded domain \(\Omega\) such that (*) has at least k solutions. This result shows that the existence and the multiplicity of solutions for the problem (*) does not depend on the topology of \(\Omega\) and suggests that the basic point is the topology of suitable nearby domains. The domains \(\Omega\) considered in this paper have a rotational symmetry with respect to one axis, which plays an important role in the proof; but other analogous existence and multiplicity results in domains without any symmetry property are contained in a paper to appear.
scientific article; zbMATH DE number 4148696
scientific article