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Liouville-type theorem for high order degenerate Lane-Emden system
Department of Mathematical Science, Tsinghua University, Beijing, 100084, China |
| $\left\{ \begin{align} & {{(-A)}^{m}}u={{v}^{p}} \\ & {{(-A)}^{m}}v={{u}^{q}}\quad \text{ in }\mathbb{R}_{+}^{n+1}:=\left\{ (x,y)|x\in {{\mathbb{R}}^{n}},y>0 \right\}, \\ & u\ge 0,v\ge 0 \\ \end{align} \right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$ |
| $ A: = y\partial_{y}^2+a\partial_{y}+\Delta_{x}, \;a\geq 1 $ |
| $ n+2a>2m, m\in \mathbb{Z}^+,\;p,\,q\geq 1 $ |
| $ 1<p,\, q<\frac{n+2a+2m}{n+2a-2m} $ |
| $ p = q = \frac{n+2a+2m}{n+2a-2m} $ |
| $ \frac{1}{p+1}+\frac{1}{q+1}>\frac{n+2a-2m}{n+2a} $ |
| $ (-A)^{m}u\geq v^p, (-A)^{m}v\geq u^q, u\geq 0, v\geq 0, $ |
| $ \mathbb{R}_+^{n+1} $ |
| $ (n+2a-2m)q<\frac{n+2a}{p}+2m $ |
| $ (n+2a-2m)p<\frac{n+2a}{q}+2m $ |
| $ p,q>1 $ |
References:
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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. |
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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. |
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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. |
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P. Clément, R. Manásevich and E. Mitidieri,
Positive solutions for a quasilinear system via blow up, Comm. Partial Differential Equations, 18 (1993), 2071-2106.
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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.
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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. |
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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. |
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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. |
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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. |
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C. Li,
Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.
doi: 10.1007/s002220050023. |
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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. 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. |
| [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. 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. |
| [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. |
| [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. Clément, R. Manásevich and E. Mitidieri,
Positive solutions for a quasilinear system via blow up, Comm. Partial Differential Equations, 18 (1993), 2071-2106.
|
| [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.
|
| [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. Han, J. 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. 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. |
| [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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