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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 |
| $Ω\subset\mathbb{R}^n$ |
| $\begin{equation*}-Δ u = μ e^{θ u +(1-θ)v} , ~~~- Δ v = λ e^{θ v + (1-θ)u}~~~\mbox{ in }Ω,\end{equation*}$ |
| $u = v = 0$ |
| $\partial Ω$ |
| $θ$ |
| $[0,1]$ |
| $μ,λ≥0$ |
| $n≤ 9$ |
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. |
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. |
| [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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