Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent

Caju, Rayssa; do Ó, João Marcos; Santos, Almir Silva

doi:10.1016/j.anihpc.2019.02.001

Caju, Rayssa ¹ ; do Ó, João Marcos ² ; Santos, Almir Silva ³

¹ Department of Mathematics, Federal University of Paraíba, 58051-900, João Pessoa-PB, Brazil
² Department of Mathematics, Brasília University, 70910-900, Brasília, DF, Brazil
³ Department of Mathematics, Federal University of Sergipe, 49100-000, São Cristovão-SE, Brazil

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 6, pp. 1575-1601.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé

We studied the asymptotic behavior of local solutions for strongly coupled critical elliptic systems near an isolated singularity. For the dimension less than or equal to five we prove that any singular solution is asymptotic to a rotationally symmetric Fowler type solution. This result generalizes the celebrated work due to Caffarelli, Gidas and Spruck [1] who studied asymptotic proprieties to the classic Yamabe equation. In addition, we generalize similar results by Marques [12] for inhomogeneous context, that is, when the metric is not necessarily conformally flat.

MR Zbl

DOI : 10.1016/j.anihpc.2019.02.001

Classification : 35J60, 35B40, 35B33, 53C21
Mots-clés : Critical equations, Elliptic systems, Asymptotic symmetry, Singular solution, A priori estimate

Affiliations des auteurs :

Caju, Rayssa ¹ ; do Ó, João Marcos ² ; Santos, Almir Silva ³

@article{AIHPC_2019__36_6_1575_0,
     author = {Caju, Rayssa and do \'O, Jo\~ao Marcos and Santos, Almir Silva},
     title = {Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1575--1601},
     publisher = {Elsevier},
     volume = {36},
     number = {6},
     year = {2019},
     doi = {10.1016/j.anihpc.2019.02.001},
     mrnumber = {4002167},
     zbl = {1425.35039},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2019.02.001/}
}

TY  - JOUR
AU  - Caju, Rayssa
AU  - do Ó, João Marcos
AU  - Santos, Almir Silva
TI  - Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2019
SP  - 1575
EP  - 1601
VL  - 36
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2019.02.001/
DO  - 10.1016/j.anihpc.2019.02.001
LA  - en
ID  - AIHPC_2019__36_6_1575_0
ER  -

%0 Journal Article
%A Caju, Rayssa
%A do Ó, João Marcos
%A Santos, Almir Silva
%T Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent
%J Annales de l'I.H.P. Analyse non linéaire
%D 2019
%P 1575-1601
%V 36
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2019.02.001/
%R 10.1016/j.anihpc.2019.02.001
%G en
%F AIHPC_2019__36_6_1575_0

Caju, Rayssa; do Ó, João Marcos; Santos, Almir Silva. Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 6, pp. 1575-1601. doi : 10.1016/j.anihpc.2019.02.001. http://www.numdam.org/articles/10.1016/j.anihpc.2019.02.001/

Bibliographie
Cité par

[1] Caffarelli, L.A.; Gidas, B.; Spruck, J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math., Volume 42 (1989), pp. 271–297 | DOI | MR | Zbl

[2] Chen, C.-C.; Lin, C.-S. Estimates of the conformal scalar curvature equation via the method of moving planes, Commun. Pure Appl. Math., Volume 50 (1997), pp. 971–1017 | MR | Zbl

[3] Chen, C.-C.; Lin, C.-S. Estimate of the conformal scalar curvature equation via the method of moving planes II, J. Differ. Geom., Volume 49 (1998), pp. 115–178 | MR | Zbl

[4] Chen, C.-C.; Lin, C.-S. On the asymptotic symmetry of singular solutions of the scalar curvature equations, Math. Ann., Volume 313 (1999), pp. 229–245 | MR | Zbl

[5] Chen, C.-C.; Lin, C.-S. Local behavior of singular positive solutions of semilinear elliptic equations with Sobolev exponent, Duke Math. J., Volume 78 (1995), pp. 315–334 | MR | Zbl

[6] Chen, Z.; Lin, C.-S. Removable singularity of positive solutions for a critical elliptic system with isolated singularity, Math. Ann., Volume 363 (2015), pp. 501–523 | DOI | MR | Zbl

[7] Druet, O.; Hebey, E. Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium, Anal. PDE, Volume 2 (2009), pp. 305–359 | DOI | MR | Zbl

[8] Druet, O.; Hebey, E.; Vétois, J. Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian, J. Funct. Anal., Volume 258 (2010), pp. 999–1059 | DOI | MR | Zbl

[9] Korevaar, N.; Mazzeo, R.; Pacard, F.; Schoen, R. Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Math., Volume 135 (1999), pp. 233–272 | DOI | MR | Zbl

[10] Lin, C.-S. Estimates of the scalar curvature equation via the method of moving planes. III, Commun. Pure Appl. Math., Volume 53 (2000), pp. 611–646 | MR | Zbl

[11] Marques, F.C. A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differ. Geom., Volume 71 (2005) no. 2, pp. 315–346 | DOI | MR | Zbl

[12] Marques, F.C. Isolated singularities of solutions to the Yamabe equation, Calc. Var. Partial Differ. Equ., Volume 32 (2008), pp. 349–371 | DOI | MR | Zbl

[13] Mazzeo, R.; Pacard, F. Constant scalar curvature metrics with isolated singularities, Duke Math. J., Volume 99 (1999), pp. 353–418 | DOI | MR | Zbl

[14] Mazzeo, R.; Pollack, D.; Uhlenbeck, K. Moduli spaces of singular Yamabe metrics, J. Am. Math. Soc., Volume 9 (1996), pp. 303–344 | DOI | MR | Zbl

[15] Reed, M.; Simon, B. Methods of Modern Mathematical Physics. IV. Analysis of Operators, 1978 (New York-London) | MR | Zbl

Cité par Sources :