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

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 4, page 559-575
- ISSN: 1292-8119

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topNazaret, Bruno. "Stability results for some nonlinear elliptic equations involving the p-Laplacian with critical Sobolev growth." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 559-575. <http://eudml.org/doc/197335>.

@article{Nazaret2010,

abstract = {
This article is devoted to the study of a perturbation with a viscosity term
in an elliptic equation involving the p-Laplacian operator and related to
the best contant problem in Sobolev inequalities in the critical case.
We prove first that this problem, together with the equation, is stable
under this perturbation, assuming some conditions on the datas. In the
next section, we show that the zero solution is strongly isolated in some
sense, among the space of the solutions. Actually, we end the paper by
giving some analoguous results in the case where the datas present
symmetries.
},

author = {Nazaret, Bruno},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Variational problems; nonlinear elliptic PDE; optimal Sobolev
inequalities; best constant problems; p-Laplacian operator.; perturbation; viscosity term; best constant problem in Sobolev inequalities},

language = {eng},

month = {3},

pages = {559-575},

publisher = {EDP Sciences},

title = {Stability results for some nonlinear elliptic equations involving the p-Laplacian with critical Sobolev growth},

url = {http://eudml.org/doc/197335},

volume = {4},

year = {2010},

}

TY - JOUR

AU - Nazaret, Bruno

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 4

SP - 559

EP - 575

AB -
This article is devoted to the study of a perturbation with a viscosity term
in an elliptic equation involving the p-Laplacian operator and related to
the best contant problem in Sobolev inequalities in the critical case.
We prove first that this problem, together with the equation, is stable
under this perturbation, assuming some conditions on the datas. In the
next section, we show that the zero solution is strongly isolated in some
sense, among the space of the solutions. Actually, we end the paper by
giving some analoguous results in the case where the datas present
symmetries.

LA - eng

KW - Variational problems; nonlinear elliptic PDE; optimal Sobolev
inequalities; best constant problems; p-Laplacian operator.; perturbation; viscosity term; best constant problem in Sobolev inequalities

UR - http://eudml.org/doc/197335

ER -

## References

top- T. Aubin, Problèmes isopérimétriques et espaces de Sobolev. J. Differential Geom.11 (1976) 573-598. Zbl0371.46011
- T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampere equations, Springer-Verlag (1982) (Grundlehren) 252. Zbl0512.53044
- 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. Zbl0955.35031
- P. Courilleau and F. Demengel, On the heat flow for p-harmonic maps with values in S1. Nonlinear Anal. TMA, accepted. Zbl0979.35082
- M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations. J. Differential Equations76 (1988) 159-189. Zbl0661.35029
- M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Analysis, Theory, Methods and Applications13 (1989) 879-902. Zbl0714.35032
- E. Hebey and M. Vaugon, Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev growth. J. Funct. Anal.119 (1994) 298-318. Zbl0798.35052
- L.C. Evans, Weak convergence methods for nonlinear partial differential equations. Conference Board of the Mathematical Sciences74 (1990).
- E. Hebey, 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. Zbl0756.35028
- E. Hebey, Sobolev Spaces on Riemannian Manifolds, Springer-Verlag (1996) (LNM) 1635. Zbl0866.58068
- A. Jourdain, Solutions nodales pour des equations de type courbure scalaire sur la sphère. Preprint of the University of Cergy-Pontoise (1997).
- P.L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part I. Revista Matematica Iberoamericana1 (1985) 145-199. Zbl0704.49005
- P.L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part II. Revista Matematica Iberoamericana1 (1985) 45-116. Zbl0704.49006
- 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. Zbl0353.46018
- P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations51 (1984) 126-150. Zbl0488.35017
- J.L. Vazquez, A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim.12 (1984) 191-202. Zbl0561.35003

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