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Nonlinear elliptic equations on Riemannian manifolds with the Sobolev critical exponent. (English) Zbl 0863.35037
Gindikin, Simon (ed.), Topics in geometry. In memory of Joseph D’Atri. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 20, 1-100 (1996).
The authors study the existence of strictly positive solutions to a semilinear problem with critical growth in the unknown function for a Schrödinger operator with bounded potential on a Riemannian \(n\)-dimensional manifold without boundary. They deal with the problem generalizing the methods used in [the first author, Critical points at infinity in some variational problems, Pitman Research Notes 182, Longman (1989; Zbl 0676.58021)] and in [the first author and J. M. Coron, Commun. Pure Appl. Math. 41, No. 3, 253-294 (1988; Zbl 0649.35033)]. Only the coercivity of the operator is assumed if \(n=3,4,5\). In the case \(n>5\) conditions on the first, second and third homology group of the manifold are also assumed.
The authors emphasize also some open problems: (1) are the conditions of homology groups also necessary to solve the problem? (2) is it possible to replace the boundedness condition on the potential by some other \(L^p\) condition?.
For the entire collection see [Zbl 0842.00040].
Reviewer: M.Biroli (Monza)

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
58J05 Elliptic equations on manifolds, general theory
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