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Radial symmetry of large solutions of nonlinear elliptic equations

Author(s): Steven D. Taliaferro
Journal: Proc. Amer. Math. Soc. 124 (1996), 447-455.
MSC (1991): Primary 35J60
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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

DOI: 10.1090/S0002-9939-96-03372-2
PII: S 0002-9939(96)03372-2
Received by editor(s): July 22, 1994
Communicated by: Jeffrey Rauch
Copyright of article: Copyright 1996, American Mathematical Society

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