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

Bruno Nazaret

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

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

Abstract

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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.

How to cite

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Nazaret, 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

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  14. P.L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part II. Revista Matematica Iberoamericana1 (1985) 45-116.  
  15. 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).  
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