Liouville-type theorems for generalized Hénon-Lane-Emden Schrödinger systems in $R^2$ and $R^3$
DOI:
https://doi.org/10.12775/TMNA.2021.028Keywords
Nonlinear elliptic weighted system, Schrödinger system, positive solutions, Liouville-type theoremAbstract
In the paper we study the Liouville-type theorems for generalized Hénon-Lane-Emden elliptic system in $\mathbb{R}^N$. By the methods of spherical averages, Rellich-Pohozaev type identities, Sobolev inequalities on $S^{N-1}$, feedback and measure arguments, and scale invariance of the solutions, we show that if the pair of exponents is subcritical, then this system has no positive solutions for $N=2$ and no bounded positive solutions for $N=3$.References
M.F. Bidaut-Véron and H. Giacomini, A new dynamical approach of Emden–Fowler equations and systems, Adv. Differential Equations 15 (2010), 1033–1082.
J. Busca and R. Manásevich, A Liouville-type theorem for Lane–Emden system, Indiana Univ. Math. J. 51 (2002), 37–51.
M. Calanchi and B. Ruf, Radial and non radial solutions for Hardy–Hénon type elliptic systems, Calc. Var. Partial Differential Equations 38 (2010), 111–133.
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 61-5-622.
W. Chen and C. Li, An integral system and the Lane–Emden conjecture, Discrete Contin. Dyn. Syst. 24 (2009), 1167–1184.
D.G. de Figueiredo and P. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super. Pisa Cl. Sci. 21 (1994), 387–397.
Y. Du, Z.Guo, Finite Morse index solutions of weighted elliptic equations and the critical exponents, Calc. Var. Partial Differential Equations 54 (2015), 3161–3181.
M. Fazly and N. Ghoussoub, On the Hénon–Lane–Emden conjecture, Discrete Contin. Dyn. Syst. 34 (2014), 2513–2533.
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), 525–598.
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer–Verlag, Berlin, 2001.
Z. Guo and F. Wan, Further study of a weighted elliptic equation, Sci. China Math. 60 (2017), 2391–2406.
K. Li and Z. Zhang, Proof of the Hénon–Lane–Emden conjecture in R3 , J. Differential Equations 266 (2019) 202–226.
K. Li and Z. Zhang, Monotonicity theorem and its applications to weighted elliptic equations, Sci. China Math. 62 (2019), 1925–1934.
C.S. Lin, A classification of solutions of a conformally invariant fourth order equation in Rn , Comment. Math. Helv. 73 (1998), 206–231.
E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations 18 (1993), 125–151.
E. Mitidieri, Non-existence of positive solutions of semilinear elliptic systems in RN , Differential Integral Equations 9 (1996), 465–479.
Q.H. Phan and P. Souplet, Liouville-type theorems and bounds of solutions of Hardy–Hénon equations, J. Differential Equations 252 (2012), 2544–2562.
P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems I. Elliptic equations and systems, Duke Math. J. 139 (2007), 555–579.
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J. 35 (1986), 681–703.
J. Serrin and H. Zou, Non-existence of positive solutions of Lane–Emden systems, Differential Integral Equations 9 (1996) , 635–653.
J. Serrin and H. Zou, Existence of positive solutions of the Lane–Emden system, Atti Semin. Mat. Fis. Univ. Modena 46 (1998) 369–380.
P. Souplet, The proof of the Lane–Emden conjecture in four space dimensions, Adv. Math. 221 (2009), 1409–1427.
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Xiyou Cheng, Kui Li, Zhitao Zhang
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
