Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations

Bidaut‐Véron, Marie‐Françoise; Верон, Лаурент

doi:10.1007/bf01243922

Cited by 275 publications

(204 citation statements)

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“…Such an inequality has been established by M.-F. Bidaut-Véron and L. Véron in [22] in the more general context of compact manifolds with uniformly positive Ricci curvature. Their method is based on the Bochner-Lichnerowicz-Weitzenböck formula and the study of the set of solutions of an elliptic equation which is seen as a bifurcation problem and contains the Euler-Lagrange equation associated to the optimality case in (1).…”

Section: Introductionmentioning

confidence: 94%

Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences

Dolbeault

Esteban²,

Kowalczyk³

et al. 2013

Chin. Ann. Math. Ser. B

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Abstract. These notes are devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension two and higher interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. We emphasize the connexion between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere. We shall address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.

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Section: Introductionmentioning

confidence: 94%

Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences

Dolbeault

Esteban²,

Kowalczyk³

et al. 2013

Chin. Ann. Math. Ser. B

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“…This type of transformation is also used by Bidaut-Véron and Véron [2], where the asymptotics of a special form of (EF ) λ has been studied. By means of (1.1), the equation (EF ) λ reduces to…”

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confidence: 99%

Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden–Fowler equations

Kristály

Rădulescu

2009

Studia Math.

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Abstract. Let (M, g) be a compact Riemannian manifold without boundary, with dim M ≥ 3, and f : R → R a continuous function which is sublinear at infinity. By various variational approaches, existence of multiple solutions of the eigenvalue problem, is established for certain eigenvalues λ > 0, depending on further properties of f and on explicit forms of the functionK. Here, ∆g stands for the Laplace-Beltrami operator on (M, g), and α,K are smooth positive functions. These multiplicity results are then applied to solve Emden-Fowler equations which involve sublinear terms at infinity.

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“…As already explained above, in the case (2, 2 * ] the above theorem was proved first in [22,Corollary 6.2], and then in [13] using previous results by E. Lieb in [34] and the Funk-Hecke formula (see [28,32]). The case p = 2 was covered in [13].…”

Section: Introductionmentioning

confidence: 72%

“…In (1), dµ is the uniform probability measure on the d-dimensional sphere, that is, the measure induced by Lebesgue's measure on S d ⊂ R d+1 , up to a normalization factor such that µ(S d ) = 1. Such an inequality has been established by M.-F. Bidaut-Véron and L. Véron in [22] in the more general context of compact manifolds with uniformly positive Ricci curvature. Their method is based on the Bochner-Lichnerowicz-Weitzenböck formula and the study of the set of solutions of an elliptic equation which is seen as a bifurcation problem and contains the Euler-Lagrange equation associated to the optimality case in (1).…”

Section: Introductionmentioning

confidence: 94%