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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[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.  Google Scholar

[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.  doi: 10.1007/BF00280741.  Google Scholar

[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.  doi: 10.1090/S0002-9939-08-09772-4.  Google Scholar

[5]

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

[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.  doi: 10.1016/j.jfa.2010.12.028.  Google Scholar

[7]

L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations,, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, (2011).  doi: 10.1201/b10802.  Google Scholar

[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, ().   Google Scholar

[9]

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

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998).   Google Scholar

[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.  doi: 10.1080/03605308008820155.  Google Scholar

[12]

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

[13]

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

[14]

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

show all references

References:
[1]

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

[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.  Google Scholar

[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.  doi: 10.1007/BF00280741.  Google Scholar

[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.  doi: 10.1090/S0002-9939-08-09772-4.  Google Scholar

[5]

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

[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.  doi: 10.1016/j.jfa.2010.12.028.  Google Scholar

[7]

L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations,, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, (2011).  doi: 10.1201/b10802.  Google Scholar

[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, ().   Google Scholar

[9]

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

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998).   Google Scholar

[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.  doi: 10.1080/03605308008820155.  Google Scholar

[12]

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

[13]

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

[14]

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

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