Rigidity results for stable solutions of symmetric systems

Fazly, Mostafa

doi:10.1090/proc/12647

Rigidity results for stable solutions of symmetric systems
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Proc. Amer. Math. Soc. 143 (2015), 5307-5321 Request permission

Abstract:

We study stable solutions of the nonlinear system \[ -\Delta u = H(u) \quad \text {in} \ \ \Omega \] where $u:\mathbb {R}^n\to \mathbb {R}^m$, $H:\mathbb {R}^m\to \mathbb {R}^m$ and $\Omega$ is a domain in $\mathbb R^n$. We introduce the novel notion of symmetric systems. The above system is said to be symmetric if the matrix of gradient of all components of $H$ is symmetric. It seems that this concept is crucial to prove Liouville theorems, when $\Omega =\mathbb R^n$, and regularity results, when $\Omega =B_1$, for stable solutions of the above system for a general nonlinearity $H \in C^1(\mathbb R ^m)$. Moreover, we provide an improvement for a linear Liouville theorem given by Fazly and Ghoussoub in 2013 that is a key tool to establish De Giorgi type results in lower dimensions for elliptic equations and systems.

References

Giovanni Alberti, Luigi Ambrosio, and Xavier Cabré, On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math. 65 (2001), no. 1-3, 9–33. Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday. MR 1843784, DOI 10.1023/A:1010602715526
Luigi Ambrosio and Xavier Cabré, Entire solutions of semilinear elliptic equations in $\mathbf R^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000), no. 4, 725–739. MR 1775735, DOI 10.1090/S0894-0347-00-00345-3
Henri Berestycki, Luis Caffarelli, and Louis Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 69–94 (1998). Dedicated to Ennio De Giorgi. MR 1655510
Haïm Brezis, Thierry Cazenave, Yvan Martel, and Arthur Ramiandrisoa, Blow up for $u_t-\Delta u=g(u)$ revisited, Adv. Differential Equations 1 (1996), no. 1, 73–90. MR 1357955
Haim Brezis and Juan Luis Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 443–469. MR 1605678
Xavier Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math. 63 (2010), no. 10, 1362–1380. MR 2681476, DOI 10.1002/cpa.20327
Xavier Cabré and Antonio Capella, On the stability of radial solutions of semilinear elliptic equations in all of $\Bbb R^n$, C. R. Math. Acad. Sci. Paris 338 (2004), no. 10, 769–774 (English, with English and French summaries). MR 2059485, DOI 10.1016/j.crma.2004.03.013
Xavier Cabré and Antonio Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal. 238 (2006), no. 2, 709–733. MR 2253739, DOI 10.1016/j.jfa.2005.12.018
Xavier Cabré and Xavier Ros-Oton, Regularity of stable solutions up to dimension 7 in domains of double revolution, Comm. Partial Differential Equations 38 (2013), no. 1, 135–154. MR 3005549, DOI 10.1080/03605302.2012.697505
Craig Cowan, Regularity of the extremal solutions in a Gelfand system problem, Adv. Nonlinear Stud. 11 (2011), no. 3, 695–700. MR 2840001, DOI 10.1515/ans-2011-0310
Craig Cowan and Mostafa Fazly, Regularity of the extremal solutions associated to elliptic systems, J. Differential Equations 257 (2014), no. 11, 4087–4107. MR 3264416, DOI 10.1016/j.jde.2014.08.002
Michael G. Crandall and Paul H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal. 58 (1975), no. 3, 207–218. MR 382848, DOI 10.1007/BF00280741
Ennio De Giorgi, Convergence problems for functionals and operators, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978) Pitagora, Bologna, 1979, pp. 131–188. MR 533166
L. Dupaigne and A. Farina, Stable solutions of $-\Delta u=f(u)$ in $\Bbb R^N$, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 855–882. MR 2654082, DOI 10.4171/JEMS/217
Louis Dupaigne, Alberto Farina, and Boyan Sirakov, Regularity of the extremal solutions for the Liouville system, Geometric partial differential equations, CRM Series, vol. 15, Ed. Norm., Pisa, 2013, pp. 139–144. MR 3156892, DOI 10.1007/978-88-7642-473-1_{7}
Pierpaolo Esposito, Nassif Ghoussoub, and Yujin Guo, Mathematical analysis of partial differential equations modeling electrostatic MEMS, Courant Lecture Notes in Mathematics, vol. 20, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010. MR 2604963, DOI 10.1090/cln/020
Pierpaolo Esposito, Nassif Ghoussoub, and Yujin Guo, Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math. 60 (2007), no. 12, 1731–1768. MR 2358647, DOI 10.1002/cpa.20189
Alberto Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\Bbb R^N$, J. Math. Pures Appl. (9) 87 (2007), no. 5, 537–561 (English, with English and French summaries). MR 2322150, DOI 10.1016/j.matpur.2007.03.001
Alberto Farina, Stable solutions of $-\Delta u=e^u$ on $\Bbb R^N$, C. R. Math. Acad. Sci. Paris 345 (2007), no. 2, 63–66 (English, with English and French summaries). MR 2343553, DOI 10.1016/j.crma.2007.05.021
Mostafa Fazly and Nassif Ghoussoub, De Giorgi type results for elliptic systems, Calc. Var. Partial Differential Equations 47 (2013), no. 3-4, 809–823. MR 3070565, DOI 10.1007/s00526-012-0536-x
Marius Ghergu and Vicenţiu D. Rădulescu, Singular elliptic problems: bifurcation and asymptotic analysis, Oxford Lecture Series in Mathematics and its Applications, vol. 37, The Clarendon Press, Oxford University Press, Oxford, 2008. MR 2488149
N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), no. 3, 481–491. MR 1637919, DOI 10.1007/s002080050196
Leon Karp, Asymptotic behavior of solutions of elliptic equations. I. Liouville-type theorems for linear and nonlinear equations on $\textbf {R}^{n}$, J. Analyse Math. 39 (1981), 75–102. MR 632457, DOI 10.1007/BF02803331
Leon Karp, Asymptotic behavior of solutions of elliptic equations. II. Analogues of Liouville’s theorem for solutions of inequalities on $\textbf {R}^{n}$, $n\geq 3$, J. Analyse Math. 39 (1981), 103–115. MR 632458, DOI 10.1007/BF02803332
Marcelo Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc. 37 (2005), no. 3, 405–416. MR 2131395, DOI 10.1112/S0024609305004248
Luisa Moschini, New Liouville theorems for linear second order degenerate elliptic equations in divergence form, Ann. Inst. H. Poincaré C Anal. Non Linéaire 22 (2005), no. 1, 11–23 (English, with English and French summaries). MR 2114409, DOI 10.1016/j.anihpc.2004.03.001
Gueorgui Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 11, 997–1002 (English, with English and French summaries). MR 1779693, DOI 10.1016/S0764-4442(00)00289-5
Salvador Villegas, Sharp estimates for semi-stable radial solutions of semilinear elliptic equations, J. Funct. Anal. 262 (2012), no. 7, 3394–3408. MR 2885956, DOI 10.1016/j.jfa.2012.01.019
Salvador Villegas, Asymptotic behavior of stable radial solutions of semilinear elliptic equations in $\Bbb R^N$, J. Math. Pures Appl. (9) 88 (2007), no. 3, 241–250 (English, with English and French summaries). MR 2355457, DOI 10.1016/j.matpur.2007.06.004
Salvador Villegas, Boundedness of extremal solutions in dimension 4, Adv. Math. 235 (2013), 126–133. MR 3010053, DOI 10.1016/j.aim.2012.11.015