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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Asymptotic behavior of the solutions of linear and quasilinear elliptic equations on $\mathbb {R}^{N}$
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by Patrick J. Rabier PDF
Trans. Amer. Math. Soc. 356 (2004), 1889-1907 Request permission

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
  • Received by editor(s): September 4, 2001
  • Received by editor(s) in revised form: August 24, 2002
  • Published electronically: October 6, 2003
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 1889-1907
  • MSC (2000): Primary 35P05, 35Q40, 47F05
  • DOI: https://doi.org/10.1090/S0002-9947-03-03234-3
  • MathSciNet review: 2031045