# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021184
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## Liouville-type theorem for high order degenerate Lane-Emden system

 Department of Mathematical Science, Tsinghua University, Beijing, 100084, China

*Corresponding author: Yuxia Guo

Revised  September 2021 Early access November 2021

Fund Project: The first author is supported by NSFC (11771235, 12031015)

In this paper, we are concerned with the following high order degenerate elliptic system:
 \left\{ \begin{align} & {{(-A)}^{m}}u={{v}^{p}} \\ & {{(-A)}^{m}}v={{u}^{q}}\quad \text{ in }\mathbb{R}_{+}^{n+1}:=\left\{ (x,y)|x\in {{\mathbb{R}}^{n}},y>0 \right\}, \\ & u\ge 0,v\ge 0 \\ \end{align} \right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)
where the operator
 $A: = y\partial_{y}^2+a\partial_{y}+\Delta_{x}, \;a\geq 1$
and
 $n+2a>2m, m\in \mathbb{Z}^+,\;p,\,q\geq 1$
. We prove the non-existence of positive smooth solutions for
 $1 , and classify positive solutions for $ p = q = \frac{n+2a+2m}{n+2a-2m} $. For $ \frac{1}{p+1}+\frac{1}{q+1}>\frac{n+2a-2m}{n+2a} $, we show the non-existence of positive, ellipse-radial, smooth solutions. Moreover we prove the non-existence of positive smooth solutions for the high order degenerate elliptic system of inequalities $ (-A)^{m}u\geq v^p, (-A)^{m}v\geq u^q, u\geq 0, v\geq 0, $in $ \mathbb{R}_+^{n+1} $for either $ (n+2a-2m)q<\frac{n+2a}{p}+2m $or $ (n+2a-2m)p<\frac{n+2a}{q}+2m $with $ p,q>1 $. Citation: Yuxia Guo, Ting Liu. Liouville-type theorem for high order degenerate Lane-Emden system. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021184 ##### References:  [1] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math, 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. Google Scholar [2] W. Chen, C. Li and B. 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Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math, 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. Google Scholar [2] W. Chen, C. Li and B. OU, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. Google Scholar [3] J. Chern and S. Yang, A divergence-type identity in a punctured domain and its applicaton to a singular polyharmonic problem, J. Dynam. Differential Equations, 16 (2004), 587-604. doi: 10.1007/s10884-004-4293-1. Google Scholar [4] P. Clément, R. Manásevich and E. Mitidieri, Positive solutions for a quasilinear system via blow up, Comm. Partial Differential Equations, 18 (1993), 2071-2106. Google Scholar [5] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in$\mathbb{R}^{n}$, Adv. Math., 7 (1981), 369-402. Google Scholar [6] Y. Guo and J. Nie, Classification for positive solutions of degenerate elliptic system, Discrete Contin. Dyn. Syst., 39 (2019), 1457-1475. doi: 10.3934/dcds.2018130. Google Scholar [7] Q. Han, J. Hong and G. Hang, Compactness of Alexandrov-Nirenberg surfaces, Comm. Pure Appl. Math., 70 (2017), 1706-1753. doi: 10.1002/cpa.21686. Google Scholar [8] G. Hang, A Liouville theorem of degenerate elliptic equation and its application, Discrete Contin. Dyn. Syst., 33 (2013), 4549-4566. doi: 10.3934/dcds.2013.33.4549. Google Scholar [9] G. Hang and C. Li, A Liouville theorem for high order degenerate elliptic equations, J. Differential Equations, 258 (2015), 1229-1251. doi: 10.1016/j.jde.2014.10.017. Google Scholar [10] G. Huang, A priori bounds for a class of semi-linear degenerate elliptic equations, Sci. China Math., 57 (2014), 1911-1926. doi: 10.1007/s11425-014-4770-x. Google Scholar [11] Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc, 6 (2004), 153-180. doi: 10.4171/JEMS/6. Google Scholar [12] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/s002220050023. Google Scholar [13] C. Lin, A classification of solutions of a conformally invariant fourth order equation in$\mathbb{R}^{n}$, Comment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052. Google Scholar [14] J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in$\mathbb{R}^N$, J. Differential Equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016. Google Scholar [15] E. Mitidieri, A Rellich-type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151. doi: 10.1080/03605309308820923. Google Scholar [16] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. ANN, 313 (1999), 207-228. doi: 10.1007/s002080050258. Google Scholar [17] X. Xu, Classification of solutions of certain fourth-order nonlinear ellliptic equations in$\mathbb{R}^4\$, Pacific J. Math., 225 (2006), 361-378.  doi: 10.2140/pjm.2006.225.361.  Google Scholar
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