June  2014, 34(6): 2495-2503. doi: 10.3934/dcds.2014.34.2495

Partial regularity for a Liouville system

1. 

Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

2. 

LAMFA, UMR CNRS 7352, Université de Picardie Jules Verne, 33 rue St Leu, 80039, Amiens Cedex, France

Received  February 2013 Revised  March 2013 Published  December 2013

Let $\Omega\subset\mathbb{R}^n$ be a bounded smooth open set. We prove that the singular set of any extremal solution of the system \begin{equation*} -\Delta u=\mu e^v , \quad - \Delta v=\lambda e^u\quad\mbox{ in }\Omega, \end{equation*} with $u=v=0$ on $\partial \Omega$, $\mu,\lambda\geq0$, has Hausdorff dimension at most $n-10$.
Citation: Juan Dávila, Olivier Goubet. Partial regularity for a Liouville system. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2495-2503. doi: 10.3934/dcds.2014.34.2495
References:
[1]

C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Adv. Nonlinear Stud., 11 (2011), 695-700.

[2]

C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains, Calc. Var. and PDEs, (2012). doi: 10.1007/s00526-012-0582-4.

[3]

M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58 (1975), 207-218. doi: 10.1007/BF00280741.

[4]

E. N. Dancer and A. Farina, On the classification of solutions of $-\Delta u=e^u$ on $\mathbb R^N$: stability outside a compact set and applications, Proc. Amer. Math. Soc., 137 (2009), 1333-1338. doi: 10.1090/S0002-9939-08-09772-4.

[5]

F. Da Lio, Partial regularity for stationary solutions to Liouville-type equation in dimension 3, Comm. Partial Differential Equations, 33 (2008), 1890-1910. doi: 10.1080/03605300802402625.

[6]

J. Dávila, L. Dupaigne and A. Farina, Partial regularity of finite Morse index solutions to the Lane-Emden equation, J. Funct. Anal., 261 (2011), 218-232. doi: 10.1016/j.jfa.2010.12.028.

[7]

L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 143, Chapman & Hall/CRC, Boca Raton, FL, 2011. doi: 10.1201/b10802.

[8]

L. Dupaigne, A. Farina and B. Sirakov, Regularity of the extremal solution for the Liouville system, to appear in Proceedings of the ERC Workshop on Geometric Partial Differential Equations, Ed. Scuola Normale Superiore di Pisa, arXiv:1207.3703.

[9]

L. Dupaigne, M. Ghergu, O. Goubet and G. Warnault, The Gel'fand for the biharmonic operator, Arch. Ration. Mech. Anal., 208 (2013), 725-752. doi: 10.1007/s00205-013-0613-0.

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[11]

F. Mignot and J.-P. Puel, Sur une classe de problèmes non linéaires avec non linéairité positive, croissante, convexe, Comm. Partial Differential Equations, 5 (1980), 791-836. doi: 10.1080/03605308008820155.

[12]

M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416. doi: 10.1112/S0024609305004248.

[13]

K. Wang, Partial regularity of stable solutions to the Emden equation, Calc. Var. Partial Differential Equations, 44 (2012), 601-610. doi: 10.1007/s00526-011-0446-3.

[14]

K. Wang, Partial regularity of stable solutions to the supercritical equations and its applications, Nonlinear Anal., 75 (2012), 5238-5260. doi: 10.1016/j.na.2012.04.041.

show all references

References:
[1]

C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Adv. Nonlinear Stud., 11 (2011), 695-700.

[2]

C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains, Calc. Var. and PDEs, (2012). doi: 10.1007/s00526-012-0582-4.

[3]

M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58 (1975), 207-218. doi: 10.1007/BF00280741.

[4]

E. N. Dancer and A. Farina, On the classification of solutions of $-\Delta u=e^u$ on $\mathbb R^N$: stability outside a compact set and applications, Proc. Amer. Math. Soc., 137 (2009), 1333-1338. doi: 10.1090/S0002-9939-08-09772-4.

[5]

F. Da Lio, Partial regularity for stationary solutions to Liouville-type equation in dimension 3, Comm. Partial Differential Equations, 33 (2008), 1890-1910. doi: 10.1080/03605300802402625.

[6]

J. Dávila, L. Dupaigne and A. Farina, Partial regularity of finite Morse index solutions to the Lane-Emden equation, J. Funct. Anal., 261 (2011), 218-232. doi: 10.1016/j.jfa.2010.12.028.

[7]

L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 143, Chapman & Hall/CRC, Boca Raton, FL, 2011. doi: 10.1201/b10802.

[8]

L. Dupaigne, A. Farina and B. Sirakov, Regularity of the extremal solution for the Liouville system, to appear in Proceedings of the ERC Workshop on Geometric Partial Differential Equations, Ed. Scuola Normale Superiore di Pisa, arXiv:1207.3703.

[9]

L. Dupaigne, M. Ghergu, O. Goubet and G. Warnault, The Gel'fand for the biharmonic operator, Arch. Ration. Mech. Anal., 208 (2013), 725-752. doi: 10.1007/s00205-013-0613-0.

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[11]

F. Mignot and J.-P. Puel, Sur une classe de problèmes non linéaires avec non linéairité positive, croissante, convexe, Comm. Partial Differential Equations, 5 (1980), 791-836. doi: 10.1080/03605308008820155.

[12]

M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416. doi: 10.1112/S0024609305004248.

[13]

K. Wang, Partial regularity of stable solutions to the Emden equation, Calc. Var. Partial Differential Equations, 44 (2012), 601-610. doi: 10.1007/s00526-011-0446-3.

[14]

K. Wang, Partial regularity of stable solutions to the supercritical equations and its applications, Nonlinear Anal., 75 (2012), 5238-5260. doi: 10.1016/j.na.2012.04.041.

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