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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some remarks on Liouville type results for quasilinear elliptic equations


Authors: E. N. Dancer and Yihong Du
Journal: Proc. Amer. Math. Soc. 131 (2003), 1891-1899
MSC (2000): Primary 35J15, 35J60
Published electronically: November 4, 2002
MathSciNet review: 1955278
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Abstract: For a wide class of nonlinearities $f(u)$satisfying

\begin{displaymath}\mbox{ $f(0)=f(a)=0$ , $f(u)>0$\space in $(0,a)$\space and $f(u)<0$\space in $(a,\infty)$ ,}\end{displaymath}

we show that any nonnegative solution of the quasilinear equation $-\Delta_p u= f(u)$ over the entire $\mathbb{R}^N$ must be a constant. Our results improve or complement some recently obtained Liouville type theorems. In particular, we completely answer a question left open by Du and Guo.


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Additional Information

E. N. Dancer
Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
Email: normd@maths.usyd.edu.au

Yihong Du
Affiliation: School of Mathematics, Statistics and Computer Science, University of New England, Armidale, New South Wales 2351, Australia
Email: ydu@turing.une.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06733-3
PII: S 0002-9939(02)06733-3
Keywords: Quasilinear elliptic equation, Liouville theorem, sweeping principle
Received by editor(s): February 8, 2002
Published electronically: November 4, 2002
Additional Notes: The work of the first author was partially supported by the Australian Research Council
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2002 American Mathematical Society