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On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain. (English) Zbl 0649.35033
This paper gives an existence result for the elliptic model problem with the critical Sobolev exponent: \(-\Delta u=u^{(N+2)/(N-2)}\) and \(u>0\) on \(\Omega\), \(u=0\) on \(\partial \Omega.\)
The authors prove, that existence is implied by \(H_ d(\Omega,{\mathbb{Z}}_ 2)\neq 0\) for some \(d\in {\mathbb{N}}\). As a corollary a solution exists in three dimensions, if \(\Omega\) is not contractible; - this was known e.g. for \(\Omega\) an annulus (and non-existence is known for \(\Omega\) starshaped). Hence this result clarifies the influence of the topology of the domain to the existence problem.
Reviewer: M.Wiegner

MSC:
35J60 Nonlinear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs
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