doi: 10.3934/dcds.2021184
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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<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 & Continuous Dynamical Systems, 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.  Google Scholar

[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.  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. ClémentR. 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. GidasW. 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. HanJ. 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. 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.  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

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.  Google Scholar

[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.  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. ClémentR. 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. GidasW. 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. HanJ. 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. 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.  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

[1]

Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317

[2]

Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure & Applied Analysis, 2016, 15 (3) : 807-830. doi: 10.3934/cpaa.2016.15.807

[3]

Quoc Hung Phan. Optimal Liouville-type theorems for a parabolic system. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 399-409. doi: 10.3934/dcds.2015.35.399

[4]

Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869

[5]

Kui Li, Zhitao Zhang. Liouville-type theorem for higher-order Hardy-Hénon system. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3851-3869. doi: 10.3934/cpaa.2021134

[6]

Dongsheng Kang, Liangshun Xu. Biharmonic systems involving multiple Rellich-type potentials and critical Rellich-Sobolev nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (2) : 333-346. doi: 10.3934/cpaa.2018019

[7]

Pavol Quittner, Philippe Souplet. Parabolic Liouville-type theorems via their elliptic counterparts. Conference Publications, 2011, 2011 (Special) : 1206-1213. doi: 10.3934/proc.2011.2011.1206

[8]

Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035

[9]

Shigeru Sakaguchi. A Liouville-type theorem for some Weingarten hypersurfaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 887-895. doi: 10.3934/dcdss.2011.4.887

[10]

Alberto Farina, Miguel Angel Navarro. Some Liouville-type results for stable solutions involving the mean curvature operator: The radial case. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 1233-1256. doi: 10.3934/dcds.2020076

[11]

Philippe Souplet. Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices. Networks & Heterogeneous Media, 2012, 7 (4) : 967-988. doi: 10.3934/nhm.2012.7.967

[12]

Foued Mtiri. Liouville type theorems for stable solutions of elliptic system involving the Grushin operator. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021187

[13]

Esa V. Vesalainen. Rellich type theorems for unbounded domains. Inverse Problems & Imaging, 2014, 8 (3) : 865-883. doi: 10.3934/ipi.2014.8.865

[14]

Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855

[15]

Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665

[16]

Hiroshi Isozaki, Hisashi Morioka. A Rellich type theorem for discrete Schrödinger operators. Inverse Problems & Imaging, 2014, 8 (2) : 475-489. doi: 10.3934/ipi.2014.8.475

[17]

Sami Baraket, Soumaya Sâanouni, Nihed Trabelsi. Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 1013-1063. doi: 10.3934/dcds.2020069

[18]

Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236

[19]

Zongming Guo, Juncheng Wei. Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2561-2580. doi: 10.3934/dcds.2014.34.2561

[20]

Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure & Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565

2020 Impact Factor: 1.392

Article outline

[Back to Top]