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Stability results for some nonlinear elliptic equations involving the p-Laplacian with critical Sobolev growth

Published online by Cambridge University Press:  15 August 2002

Bruno Nazaret
Bruno Nazaret*
Affiliation:
ENS Cachan, Antenne de Bretagne, Campus de Ker Lann, 35170 Bruz, France. Université de Cergy-Pontoise, Département de Mathématiques, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France.
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Abstract

This article is devoted to the study of a perturbation with a viscosity termin an elliptic equation involving the p-Laplacian operator and related tothe best contant problem in Sobolev inequalities in the critical case.We prove first that this problem, together with the equation, is stableunder this perturbation, assuming some conditions on the datas. In thenext section, we show that the zero solution is strongly isolated in somesense, among the space of the solutions. Actually, we end the paper bygiving some analoguous results in the case where the datas presentsymmetries.

Keywords

Variational problemsnonlinear elliptic PDEoptimal Sobolevinequalitiesbest constant problemsp-Laplacian operator.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 559 - 575
Copyright
© EDP Sciences, SMAI, 1999

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References

Aubin, T., Problèmes isopérimétriques et espaces de Sobolev. J. Differential Geom. 11 (1976) 573-598. CrossRef
T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampere equations, Springer-Verlag (1982) (Grundlehren) 252.
O. Druet, Generalized scalar curvature type equations on compact riemaniann manifolds. Preprint of the University of Cergy-Pontoise (1997).
F. Demengel and E. Hebey, On some nonlinear equations involving the p-Laplacian with critical Sobolev growth. Adv. in PDE's, to appear.
P. Courilleau and F. Demengel, On the heat flow for p-harmonic maps with values in S1 . Nonlinear Anal. TMA, accepted.
Guedda, M. and Veron, L., Local and global properties of solutions of quasilinear elliptic equations. J. Differential Equations 76 (1988) 159-189. CrossRef
M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Analysis, Theory, Methods and Applications 13 (1989) 879-902.
Hebey, E. and Vaugon, M., Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev growth. J. Funct. Anal. 119 (1994) 298-318. CrossRef
L.C. Evans, Weak convergence methods for nonlinear partial differential equations. Conference Board of the Mathematical Sciences 74 (1990).
Hebey, E., La méthode d'isométries-concentration dans le cas d'un problème non linéaire sur les variétés compactes à bord avec exposant critique de sobolev. Bull. Sci. Math. 116 (1992) 35-51.
E. Hebey, Sobolev Spaces on Riemannian Manifolds, Springer-Verlag (1996) (LNM) 1635.
A. Jourdain, Solutions nodales pour des equations de type courbure scalaire sur la sphère. Preprint of the University of Cergy-Pontoise (1997).
Lions, P.L., The concentration-compactness principle in the calculus of variations. The limit case, part I. Revista Matematica Iberoamericana 1 (1985) 145-199. CrossRef
Lions, P.L., The concentration-compactness principle in the calculus of variations. The limit case, part II. Revista Matematica Iberoamericana 1 (1985) 45-116. CrossRef
B. Nazaret, Stabilité sous des perturbations visqueuses des solutions d'équations du type p-Laplacien avec exposant critique de Sobolev. Preprint of the University of Cergy-Pontoise (5/98).
G. Talenti. Best constants in Sobolev inequalities. Ann. Mat. Pura Appl. 110 (1976) 353-372.
Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations 51 (1984) 126-150. CrossRef
Vazquez, J.L., A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12 (1984) 191-202. CrossRef