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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Asymptotic behavior of the solutions of linear and quasilinear elliptic equations on $\mathbb{R} ^{N}$


Author: Patrick J. Rabier
Journal: Trans. Amer. Math. Soc. 356 (2004), 1889-1907
MSC (2000): Primary 35P05, 35Q40, 47F05
Published electronically: October 6, 2003
MathSciNet review: 2031045
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Abstract: We investigate the relationship between the decay at infinity of the right-hand side $f$ and solutions $u$ of an equation $Lu=f$ when $L$ is a second order elliptic operator on $\mathbb{R} ^{N}.$ It is shown that when $L$is Fredholm, $u$ inherits the type of decay of $f$ (for instance, exponential, or power-like). In particular, the generalized eigenfunctions associated with all the Fredholm eigenvalues of $L,$ isolated or not, decay exponentially. No use is made of spectral theory. The result is next extended when $L$ is replaced by a Fredholm quasilinear operator. Various generalizations to other unbounded domains, higher order operators or elliptic systems are possible and briefly alluded to, but not discussed in detail.


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Additional Information

Patrick J. Rabier
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: rabier@imap.pitt.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03234-3
PII: S 0002-9947(03)03234-3
Keywords: Fredholm operator, unique continuation, eigenvalue, generalized eigenfunction, exponential decay.
Received by editor(s): September 4, 2001
Received by editor(s) in revised form: August 24, 2002
Published electronically: October 6, 2003
Article copyright: © Copyright 2003 American Mathematical Society