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Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains. (English) Zbl 0701.35068

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.
Reviewer: D.Passaseo

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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References:

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