Volume 18, Issue 2
Liouville Type Theorems of Semilinear Equations with Square Sum of Vector Fields

Yazhou Han , Xuebo Luo & Pengcheng Niu

J. Part. Diff. Eq., 18 (2005), pp. 149-153.

Published online: 2005-05

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  • Abstract

Let X_j ; j = 1, …, k, be first order smooth quasi-homogeneous vector fields on Rn with the property that the dimension of the Lie algebra generated by these vector fields is n at x = 0 and X^∗_j = -X_j, j = 1, …, k. Let L = \sum^k_{i=1} X²_i . In this paper, we study the nonnegative solutions of semilinear equation Lu + f(x, u) = 0 (or ≤ 0 ) in Rn and generalized cone domain, respectively, and prove that the solutions must be vanish under some suitable conditions.

  • Keywords

Liouville type theorem superlinear equation local Hörmander condition square sum operator generalized cone domain

  • AMS Subject Headings

35B05 35H99.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-18-149, author = {}, title = {Liouville Type Theorems of Semilinear Equations with Square Sum of Vector Fields}, journal = {Journal of Partial Differential Equations}, year = {2005}, volume = {18}, number = {2}, pages = {149--153}, abstract = { Let X_j ; j = 1, …, k, be first order smooth quasi-homogeneous vector fields on Rn with the property that the dimension of the Lie algebra generated by these vector fields is n at x = 0 and X^∗_j = -X_j, j = 1, …, k. Let L = \sum^k_{i=1} X²_i . In this paper, we study the nonnegative solutions of semilinear equation Lu + f(x, u) = 0 (or ≤ 0 ) in Rn and generalized cone domain, respectively, and prove that the solutions must be vanish under some suitable conditions.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5351.html} }
TY - JOUR T1 - Liouville Type Theorems of Semilinear Equations with Square Sum of Vector Fields JO - Journal of Partial Differential Equations VL - 2 SP - 149 EP - 153 PY - 2005 DA - 2005/05 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5351.html KW - Liouville type theorem KW - superlinear equation KW - local Hörmander condition KW - square sum operator KW - generalized cone domain AB - Let X_j ; j = 1, …, k, be first order smooth quasi-homogeneous vector fields on Rn with the property that the dimension of the Lie algebra generated by these vector fields is n at x = 0 and X^∗_j = -X_j, j = 1, …, k. Let L = \sum^k_{i=1} X²_i . In this paper, we study the nonnegative solutions of semilinear equation Lu + f(x, u) = 0 (or ≤ 0 ) in Rn and generalized cone domain, respectively, and prove that the solutions must be vanish under some suitable conditions.
Yazhou Han , Xuebo Luo & Pengcheng Niu . (2019). Liouville Type Theorems of Semilinear Equations with Square Sum of Vector Fields. Journal of Partial Differential Equations. 18 (2). 149-153. doi:
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