Comptes Rendus
Partial Differential Equations
Capacitary estimates of solutions of a class of nonlinear elliptic equations
[Estimations capacitaires des solutions d'une classe d'équations elliptiques non linéaires]
Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 913-918.

Soit Ω un domaine borné régulier de N and K un sous-ensemble compact de Ω. Supposons q⩾(N+1)/(N−1) et soit UK la solution maximale de ()-Δu+u q =0 dans Ω qui s'annulle sur ΩK. Nous obtenons des majorations et minorations précises de UK au moyen de la capacité de Bessel C2/q,q et montrons que UK est σ-modérée. En outre nous corrélons les points d'explosion forte de UK et les points épais de K pour la topologie fine associée à C2/q,q et caractérisons ces points par une condition d'intégrale de chemin portant sur UK.

Let Ω be a smooth bounded domain in N and K a compact subset of Ω. Assume that q⩾(N+1)/(N−1) and denote by UK the maximal solution of −Δu+uq=0 in Ω which vanishes on ΩK. We obtain sharp upper and lower estimates for UK in terms of the Bessel capacity C2/q,q and prove that UK is σ-moderate. In addition we relate the strong ‘blow-up’ points of UK on Ω to the ‘thick’ points of K in the fine topology associated with C2/q,q and characterize these points by a path integral condition on UK.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00217-6

Moshe Marcus 1 ; Laurent Véron 2

1 Department of Mathematics, Israel Institute of Technology-Technion, 32000 Haifa, Israel
2 Département de mathématiques, Faculté des sciences et techniques, Université de Tours, 37200 Tours, France
@article{CRMATH_2003__336_11_913_0,
     author = {Moshe Marcus and Laurent V\'eron},
     title = {Capacitary estimates of solutions of a class of nonlinear elliptic equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {913--918},
     publisher = {Elsevier},
     volume = {336},
     number = {11},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00217-6},
     language = {en},
}
TY  - JOUR
AU  - Moshe Marcus
AU  - Laurent Véron
TI  - Capacitary estimates of solutions of a class of nonlinear elliptic equations
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 913
EP  - 918
VL  - 336
IS  - 11
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00217-6
LA  - en
ID  - CRMATH_2003__336_11_913_0
ER  - 
%0 Journal Article
%A Moshe Marcus
%A Laurent Véron
%T Capacitary estimates of solutions of a class of nonlinear elliptic equations
%J Comptes Rendus. Mathématique
%D 2003
%P 913-918
%V 336
%N 11
%I Elsevier
%R 10.1016/S1631-073X(03)00217-6
%G en
%F CRMATH_2003__336_11_913_0
Moshe Marcus; Laurent Véron. Capacitary estimates of solutions of a class of nonlinear elliptic equations. Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 913-918. doi : 10.1016/S1631-073X(03)00217-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00217-6/

[1] D.R. Adams; L.I. Hedberg Function Spaces and Potential Theory, Grundlehren Math. Wiss., 314, Springer, 1996

[2] E.B. Dynkin; S.E. Kuznetsov Superdiffusions and removable singularities for quasilinear partial differential equations, Comm. Pure Appl. Math., Volume 49 (1996), pp. 125-176

[3] E.B. Dynkin; S.E. Kuznetsov Solutions of Lu=uα dominated by harmonic functions, J. Analyse Math., Volume 68 (1996), pp. 15-37

[4] D.A. Labutin, Wiener regularity for large solutions of nonlinear equations, Arch. Math., à paraı̂tre

[5] J.F. Legall The Brownian snake and solutions of Δu=u2 in a domain, Probab. Theory Related Fields, Volume 102 (1995), pp. 393-432

[6] M. Marcus; L. Véron The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case, Arch. Rational Mech. Anal., Volume 144 (1998), pp. 201-231

[7] M. Marcus; L. Véron The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case, J. Math. Pures Appl., Volume 77 (1998), pp. 481-524

[8] M. Marcus; L. Véron Removable singularities and boundary trace, J. Math. Pures Appl., Volume 80 (2000), pp. 879-900

[9] B. Mselati, Classification et représentation probabiliste des solutions positives de Δu=u2 dans un domaine, Thèse de Doctorat, Université Paris 6, 2002

Cité par Sources :

This research was supported by RTN contract No. HPRN-CT-2002-00274.

Commentaires - Politique