Radial symmetry of large solutions of nonlinear elliptic equations

Taliaferro, Steven

doi:10.1090/S0002-9939-96-03372-2

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: 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 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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