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Positive solutions for some semilinear elliptic equations with critical Sobolev exponents. (English) Zbl 0635.35033
This paper deals with positive solutions of nonlinear Poisson equations with Laplacian on the left-hand side and a nonlinear nonautonomous function on the right-hand side such as arise in conformal geometry and physics. In particular attention is focused on the existence theory when the domain is a manifold with boundary and the Laplacian is given by the Riemannian metric. The results generalize and extend in various directions those of Brezis-Nirenberg for flat manifolds, and the manifolds are inspired by the variational methods of Brezis-Nirenberg and the work of Schoen. The paper is clear given the highly technical nature of some of the calculations.
Reviewer: J.F.Toland

35J65 Nonlinear boundary value problems for linear elliptic equations
32J99 Compact analytic spaces
53A30 Conformal differential geometry (MSC2010)
35A15 Variational methods applied to PDEs
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI
[1] Aubin, J. Diff. Geom. 11 pp 573– (1976)
[2] Brezis, Comm. on Pure and App. Math. 36 pp 437– (1983)
[3] Escobar, Inventiones Mathematicae
[4] Pohozaev, Soviet Math. Doklady 6 pp 1408– (1985)
[5] Translated from the Russian Dokl. Akad. Nauk SSSR 165, 1965, pp. 33–36.
[6] Schoen, J. Diff. Geom. 20 pp 479– (1984)
[7] Talenti, Ann. di Matematica, Ser. 4 pp 110– (1976)
[8] Trudinger, Ann. Scu. Norm. Sup. Pisa 22 pp 265– (1968)
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