May  2022, 42(5): 2073-2100. doi: 10.3934/dcds.2021184

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 Published  May 2022 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<p,\, q<\frac{n+2a+2m}{n+2a-2m} $
, 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 and Continuous Dynamical Systems, 2022, 42 (5) : 2073-2100. doi: 10.3934/dcds.2021184
References:
[1]

L. CaffarelliB. 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.

[2]

W. ChenC. Li and B. OU, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[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.

[4]

P. ClémentR. Manásevich and E. Mitidieri, Positive solutions for a quasilinear system via blow up, Comm. Partial Differential Equations, 18 (1993), 2071-2106. 

[5]

B. GidasW. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{n}$, Adv. Math., 7 (1981), 369-402. 

[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.

[7]

Q. HanJ. Hong and G. Hang, Compactness of Alexandrov-Nirenberg surfaces, Comm. Pure Appl. Math., 70 (2017), 1706-1753.  doi: 10.1002/cpa.21686.

[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.

[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.

[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.

[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.

[12]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023.

[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.

[14]

J. LiuY. 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.

[15]

E. Mitidieri, A Rellich-type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.  doi: 10.1080/03605309308820923.

[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.

[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.

show all references

References:
[1]

L. CaffarelliB. 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.

[2]

W. ChenC. Li and B. OU, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[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.

[4]

P. ClémentR. Manásevich and E. Mitidieri, Positive solutions for a quasilinear system via blow up, Comm. Partial Differential Equations, 18 (1993), 2071-2106. 

[5]

B. GidasW. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{n}$, Adv. Math., 7 (1981), 369-402. 

[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.

[7]

Q. HanJ. Hong and G. Hang, Compactness of Alexandrov-Nirenberg surfaces, Comm. Pure Appl. Math., 70 (2017), 1706-1753.  doi: 10.1002/cpa.21686.

[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.

[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.

[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.

[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.

[12]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023.

[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.

[14]

J. LiuY. 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.

[15]

E. Mitidieri, A Rellich-type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.  doi: 10.1080/03605309308820923.

[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.

[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.

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