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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

MSC:
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
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References:
[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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