no. 6
Partial differential equations
Stability for entire radial solutions to the biharmonic equation with negative exponents
[Stabilité des solutions radiales entières de l'équation biharmonique avec exposants négatifs]

Présenté par : Haïm Brézis

Huang, Xia1 ; Ye, Dong2 ; Zhou, Feng1
1 Center for Partial Differential Equations, School of Mathematical Sciences, East China Normal University, Shanghai, 200241, China
2 IECL, UMR 7502, Département de mathématiques, Université de Lorraine, 3, rue Augustin-Fresnel, 57073 Metz, France
Comptes Rendus. Mathématique, Tome 356 (2018) no. 6, pp. 632-636.

Dans cette note, on s'intéresse aux solutions radiales entières de l'équation semilinéaire biharmonique

Δ2u=−u−p,u>0dansRN,
où p>0 et N≥3. En particulier, on étudie la stabilité en dehors d'un compact des solutions radiales entières, et on résout un cas ouvert dans [5].

In this note, we are interested in entire solutions to the semilinear biharmonic equation

Δ2u=−u−p,u>0inRN,
where p>0 and N≥3. In particular, the stability outside a compact set of the entire radial solutions will be completely studied, which resolves the remaining case in [5].

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.05.001
Huang, Xia 1 ; Ye, Dong 2 ; Zhou, Feng 1

1 Center for Partial Differential Equations, School of Mathematical Sciences, East China Normal University, Shanghai, 200241, China
2 IECL, UMR 7502, Département de mathématiques, Université de Lorraine, 3, rue Augustin-Fresnel, 57073 Metz, France 
@article{CRMATH_2018__356_6_632_0,
     author = {Huang, Xia and Ye, Dong and Zhou, Feng},
     title = {Stability for entire radial solutions to the biharmonic equation with negative exponents},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {632--636},
     publisher = {Elsevier},
     volume = {356},
     number = {6},
     year = {2018},
     doi = {10.1016/j.crma.2018.05.001},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2018.05.001/}
}
TY  - JOUR
AU  - Huang, Xia
AU  - Ye, Dong
AU  - Zhou, Feng
TI  - Stability for entire radial solutions to the biharmonic equation with negative exponents
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 632
EP  - 636
VL  - 356
IS  - 6
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2018.05.001/
DO  - 10.1016/j.crma.2018.05.001
LA  - en
ID  - CRMATH_2018__356_6_632_0
ER  - 
%0 Journal Article
%A Huang, Xia
%A Ye, Dong
%A Zhou, Feng
%T Stability for entire radial solutions to the biharmonic equation with negative exponents
%J Comptes Rendus. Mathématique
%D 2018
%P 632-636
%V 356
%N 6
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2018.05.001/
%R 10.1016/j.crma.2018.05.001
%G en
%F CRMATH_2018__356_6_632_0
Huang, Xia; Ye, Dong; Zhou, Feng. Stability for entire radial solutions to the biharmonic equation with negative exponents. Comptes Rendus. Mathématique, Tome 356 (2018) no. 6, pp. 632-636. doi : 10.1016/j.crma.2018.05.001. http://archive.numdam.org/articles/10.1016/j.crma.2018.05.001/

[1] Caldiroli, P.; Musina, R. Rellich inequalities with weights, Calc. Var. Partial Differ. Equ., Volume 45 (2012) no. 1–2, pp. 147-164

[2] Dávila, J.; Flores, I.; Guerra, I. Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., Volume 348 (2010) no. 1, pp. 143-193

[3] Guerra, I. A note on nonlinear biharmonic equations with negative exponents, J. Differ. Equ., Volume 253 (2012) no. 11, pp. 3147-3157

[4] Kusano, T.; Naita, M.; Swanson, C.A. Radial entire solutions of even order semilinear elliptic equations, Can. J. Math., Volume 40 (1988), pp. 1281-1300

[5] Lai, B.S. The regularity and stability of solutions to semilinear fourth-order elliptic problems with negative exponents, Proc. R. Soc. Edinb., Sect. A, Volume 146 (2016) no. 1, pp. 195-212

[6] Lai, B.S.; Ye, D. Remarks on entire solutions for two fourth-order elliptic problems, Proc. Edinb. Math. Soc., Volume 59 (2016) no. 3, pp. 777-786

[7] McKenna, P.J.; Reichel, W. Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry, Electron. J. Differ. Equ., Volume 2003 (2003) no. 37 (13 p)

[8] Warnault, G. Liouville theorems for stable radial solutions for the biharmonic operator, Asymptot. Anal., Volume 69 (2010), pp. 87-98

Cité par Sources :