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Generalized scalar curvature type equations on compact Riemannian manifolds

✍ Scribed by Olivier Druet

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
306 KB
Volume
327
Category
Article
ISSN
0764-4442

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✦ Synopsis

Let (M, g) be a compact Riemannian manifold of dimension n ~2. First, we study the existence of positive solutions for generalized scalar curvature type equations; these equations are nonlinear, of critical Sobolev growth, and involve the p-Laplacian. Then, we study similar type of equations, but in the presence of a perturbation. © Academic des ScienceslElsevier, Paris Equation de type courbure scalaire gelleralisee sur des varietes riemanniennes compactes Resume. Soit (M, g) une variete riemannienne compacte de dimension n ~2. On etudle dans un premier temps I'existence de solutions positives pour des equations de type courbure scalaire generalisee ; res equations sont non lineaires. acroissance de Sobolev critique, et font intervenir Ie p-Iaplacien. Dans un deuxieme temps on etudie Ie meme type d 'equations. mais en presence d 'une perturbation. © Academic des Sciences/Elsevier, Paris Version franraise abregee Soit (M,g) une variete riemannienne compacte de dimension n ~2 et p E (1, n). Pour a, ! E COO(M, R), on etudie I'existence de solutions positives u E Hf(M) (I'espace de Sobolev standard d'ordre p) a l'equation : ~u + au p -1 -!u P ' -l p -, (E) OU p* = ;f!p et ~pu = -div g (!'\7ul p -2 '\7u) est Ie p-Iaplacien de u.

L'equation (E), qui est une extension de l'equation de courbure scalaire prescrite, est dite de « type courbure scalaire generalisee », Dans l'etude de cette equation, l'operateur Lgu = ~pu + alu!p-2u joue un role fondamental.

Note presentee par Thierry AUBIN.

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