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Proceedings of the American Mathematical Society

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

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Radial symmetry of large solutions of nonlinear elliptic equations
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by Steven D. Taliaferro PDF
Proc. Amer. Math. Soc. 124 (1996), 447-455 Request permission

Abstract:

We give conditions under which all $C^2$ solutions of the problem \begin{align*} &\Delta u = f(|x|,u),\qquad x\in {\mathbb {R}}^n,\ &\lim _{|x|\to \infty } u(x) = \infty \end{align*} are radial. We assume $f(|x|,u)$ is positive when $|x|$ and $u$ are both large and positive. Since this problem with $f(|x|,u) = u$ has non-radial solutions, we rule out this possibility by requiring that $f(|x|,u)$ grow superlinearly in $u$ when $|x|$ and $u$ are both large and positive. However we make no assumptions on the rate of growth of solutions.
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Additional Information
  • Steven D. Taliaferro
  • Affiliation: Mathematics Department, Texas A&M University, College Station, Texas 77843
  • Email: stalia@math.tamu.edu
  • Received by editor(s): July 22, 1994
  • Communicated by: Jeffrey Rauch
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 447-455
  • MSC (1991): Primary 35J60
  • DOI: https://doi.org/10.1090/S0002-9939-96-03372-2
  • MathSciNet review: 1327049