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

Regularity of extremal solutions of a Liouville system

Olivier Goubet

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
$n≤ 9$
.
Keywords: Regularity, Liouville system, extremal solution.
Mathematics Subject Classification: Primary: 35G30, 35B65; Secondary: 35P30.
Citation: Olivier Goubet. Regularity of extremal solutions of a Liouville system. Discrete & 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.  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, 49 (2014), 291-305.  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]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

show all references

References:
[1]

C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Adv. Nonlinear Stud., 11 (2011), 695-700.  doi: 10.1515/ans-2011-0310.  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, 49 (2014), 291-305.  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]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

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