A Liouville type theorem for some critical semilinear elliptic equations on noncompact manifolds

2001 ◽  
Vol 50 (4) ◽  
pp. 1915-1936 ◽  
Author(s):  
Qi S. Zhang
Keyword(s):  
Elliptic Equations ◽  
Type Theorem ◽  
Semilinear Elliptic Equations ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Noncompact Manifolds ◽  
Semilinear Elliptic

The polynomial growth solutions to some sub-elliptic equations on the Heisenberg group

2019 ◽  
Vol 21 (01) ◽  
pp. 1750069 ◽  
Author(s):  
Hairong Liu ◽  
Tian Long ◽  
Xiaoping Yang
Keyword(s):  
Dirichlet Problem ◽  
Heisenberg Group ◽  
Elliptic Equations ◽  
Polynomial Growth ◽  
Bounded Solution ◽  
Type Theorem ◽  
Divergence Form ◽  
Liouville Type ◽  
Periodic Coefficients ◽  
Liouville Type Theorem

We give an explicit description of polynomial growth solutions to some sub-elliptic operators of divergence form with [Formula: see text]-periodic coefficients on the Heisenberg group, where the periodicity has to be meant with respect to the Heisenberg geometry. We show that the polynomial growth solutions are necessarily polynomials with [Formula: see text]-periodic coefficients. We also prove the Liouville-type theorem for the Dirichlet problem to these sub-elliptic equations on an unbounded domain on the Heisenberg group, show that any bounded solution to the Dirichlet problem must be constant.

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