Positive solutions for some semilinear elliptic equations with critical Sobolev exponents.

*(English)*Zbl 0635.35033This 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 |

##### Keywords:

semilinear elliptic equations; Sobolev exponent; positive solutions; nonlinear Poisson equations; conformal geometry; existence; manifold with boundary; Riemannian metric; flat manifolds; variational methods
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\textit{J. F. Escobar}, Commun. Pure Appl. Math. 40, No. 5, 623--657 (1987; Zbl 0635.35033)

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##### References:

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[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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