Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

A new critical curve for the Lane-Emden system
Pages: 2469 - 2479, Issue 6, June 2014

doi:10.3934/dcds.2014.34.2469      Abstract        References        Full text (394.2K)           Related Articles

Wenjing Chen - Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile (email)
Louis Dupaigne - Institut Camille Jordan UMR CNRS 5208, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France (email)
Marius Ghergu - School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland (email)

1 J. Busca and B. Sirakov, Harnack type estimates for nonlinear elliptic systems and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 543-590.       
2 P. Caldiroli and R. Musina, Rellich inequalities with weights, Calc. Var. Partial Differential Equations, 45 (2012), 147-164.       
3 C. Cowan, Regularity of stable solutions of a Lane-Emden type system, preprint, arXiv:1206.4273.
4 C. Cowan, Liouville theorems for stable Lane-Emden systems and biharmonic problems, Nonlinearity, 26 (2003), 2357-2371.
5 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.       
6 J. Dávila, L. Dupaigne and M. Montenegro, The extremal solution of a boundary reaction problem, Commun. Pure Appl. Anal., 7 (2008), 795-817.       
7 S. Dumont, L. Dupaigne, O. Goubet and V. Rădulescu, Back to the Keller-Osserman condition for boundary blow-up solutions, Adv. Nonlinear Stud., 7 (2007), 271-298.       
8 L. Dupaigne, M. Ghergu, O. Goubet and G. Warnault, The Gel'fand problem for the biharmonic operator, Arch. Ration. Mech. Anal., 208 (2013), 725-752.       
9 A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbbR^N$, J. Math. Pures Appl. (9), 87 (2007), 537-561.       
10 H. Hajlaoui, A. Harrabi and D. Ye, On stable solutions of biharmonic problem with polynomial growth, preprint, arXiv:1211.2223.
11 D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.       
12 F. Gazzola and H.-C. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936.       
13 P. Karageorgis, Stability and intersection properties of solutions to the nonlinear biharmonic equation, Nonlinearity, 22 (2009), 1653-1661.       
14 E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.       
15 E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$, Differential Integral Equations, 9 (1996), 465-479.       
16 M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.       
17 P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.       
18 J. Serrin and H. Zou, Non-existence of positive solutions of semilinear elliptic systems, in A tribute to Ilya Bakelman (eds. I. R. Bakelman, S. A. Fulling and S. D. Taliaferro) (College Station, TX, 1993), Discourses Math. Appl., 3, Texas A & M Univ., College Station, TX, 1994, 55-68.       
19 Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.       
20 G. Sweers, Strong positivity in $C(\overline\Omega)$ for elliptic systems, Math. Z., 209 (1992), 251-271.       
21 R. C. A. M Van der Vorst, Variational identities and applications to differential systems, Arch. Rational Mech. Anal., 116 (1992), 375-398.       
22 J. Wei, X. Xu and Y. Wen, On the classification of stable solutions to biharmonic problems in large dimensions, Pacific J. Math., 263 (2013), 495-512.       
23 J. Wei and D. Ye, Liouville theorems for stable solutions of biharmonic problem, Math. Ann., 356 (2013), 1599-1612.       

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