Transactions of the American Mathematical Society

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

Author(s): Patrick J. Rabier
Journal: Trans. Amer. Math. Soc. 356 (2004), 1889-1907.
MSC (2000): Primary 35P05, 35Q40, 47F05
Posted: October 6, 2003
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Abstract: We investigate the relationship between the decay at infinity of the right-hand side

and solutions

of an equation

when

is a second order elliptic operator on $\mathbb{R} ^{N}.$ It is shown that when

is Fredholm,

inherits the type of decay of

(for instance, exponential, or power-like). In particular, the generalized eigenfunctions associated with all the Fredholm eigenvalues of

isolated or not, decay exponentially. No use is made of spectral theory. The result is next extended when

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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Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35P05, 35Q40, 47F05

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

DOI: 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
Posted: October 6, 2003
Copyright of article: Copyright 2003, American Mathematical Society

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