Radial symmetry of large solutions of nonlinear elliptic equations

Author:
Steven D. Taliaferro

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

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Abstract | References | Similar Articles | Additional Information

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

Article copyright:
© Copyright 1996
American Mathematical Society