Radial symmetry of large solutions of nonlinear elliptic equations

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(...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 1996-02, Vol.124 (2), p.447-455
Main Author: Taliaferro, Steven D.
Format: Article
Language:English
Subjects:
Elliptic equations
Mathematical procedures
Mathematical theorems
Maximum principle
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Summary: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.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-96-03372-2