# American Institute of Mathematical Sciences

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, 2014, 34 (6) : 2495-2503. doi: 10.3934/dcds.2014.34.2495
##### References:
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##### References:
 [1] C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Adv. Nonlinear Stud., 11 (2011), 695-700.  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), 207-218. 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), 1333-1338. 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), 1890-1910. 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), 218-232. 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, 143, Chapman & Hall/CRC, Boca Raton, FL, 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), 725-752. 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, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  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), 791-836. doi: 10.1080/03605308008820155.  Google Scholar [12] M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416. 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), 601-610. 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), 5238-5260. doi: 10.1016/j.na.2012.04.041.  Google Scholar
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