# American Institute of Mathematical Sciences

April  2019, 12(2): 339-345. doi: 10.3934/dcdss.2019023

## Regularity of extremal solutions of a Liouville system

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

Received  May 2017 Revised  December 2017 Published  August 2018

Let
 $Ω\subset\mathbb{R}^n$
be a bounded smooth open set. We prove that the extremal solution of the system
 $\begin{equation*}-Δ u = μ e^{θ u +(1-θ)v} , ~~~- Δ v = λ e^{θ v + (1-θ)u}~~~\mbox{ in }Ω,\end{equation*}$
with
 $u = v = 0$
on
 $\partial Ω$
,
 $θ$
in
 $[0,1]$
and
 $μ,λ≥0$
are smooth if
 $n≤ 9$
.
Citation: Olivier Goubet. Regularity of extremal solutions of a Liouville system. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 339-345. doi: 10.3934/dcdss.2019023
##### References:
 [1] C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Adv. Nonlinear Stud., 11 (2011), 695-700.  doi: 10.1515/ans-2011-0310. [2] C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains, Calc. Var. and PDEs, 49 (2014), 291-305.  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] 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. [5] J. Dávila and O. Goubet, Partial regularity for a Liouville system, Discrete Contin. Dyn. Syst., 34 (2014), 2495-2503.  doi: 10.3934/dcds.2014.34.2495. [6] 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. [7] L. Dupaigne, A. Farina and B. Sirakov, Regularity of the extremal solution for the Liouville system, Geometric Partial Differential Equations, 139-144, CRM Series, 15, Ed. Norm., Pisa, 2013. arXiv: 1207.3703. doi: 10.1007/978-88-7642-473-1_7. [8] 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. [9] A. Farina, On the classification of solutions of the solutions of Lane-Emden equations on unbounded domains of $\mathbb{R}^n$, J. Math. Pures et Appliquées, 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001. [10] 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. [11] M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.  doi: 10.1112/S0024609305004248. [12] 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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##### References:
 [1] C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Adv. Nonlinear Stud., 11 (2011), 695-700.  doi: 10.1515/ans-2011-0310. [2] C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains, Calc. Var. and PDEs, 49 (2014), 291-305.  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] 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. [5] J. Dávila and O. Goubet, Partial regularity for a Liouville system, Discrete Contin. Dyn. Syst., 34 (2014), 2495-2503.  doi: 10.3934/dcds.2014.34.2495. [6] 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. [7] L. Dupaigne, A. Farina and B. Sirakov, Regularity of the extremal solution for the Liouville system, Geometric Partial Differential Equations, 139-144, CRM Series, 15, Ed. Norm., Pisa, 2013. arXiv: 1207.3703. doi: 10.1007/978-88-7642-473-1_7. [8] 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. [9] A. Farina, On the classification of solutions of the solutions of Lane-Emden equations on unbounded domains of $\mathbb{R}^n$, J. Math. Pures et Appliquées, 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001. [10] 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. [11] M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.  doi: 10.1112/S0024609305004248. [12] 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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