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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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