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.

CL W. Chen and C. Li, Classification of solutions to a singular nonlinear elliptic equation, preprint.
GNN B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in 𝑅ⁿ, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 369–402. MR 634248
LN1 Yi Li and Wei-Ming Ni, On the asymptotic behavior and radial symmetry of positive solutions of semilinear elliptic equations in 𝑅ⁿ. I. Asymptotic behavior, Arch. Rational Mech. Anal. 118 (1992), no. 3, 195–222. MR 1158935, https://doi.org/10.1007/BF00387895
LN2 Yi Li and Wei-Ming Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in 𝑅ⁿ, Comm. Partial Differential Equations 18 (1993), no. 5-6, 1043–1054. MR 1218528, https://doi.org/10.1080/03605309308820960
T S.D. Taliaferro, Are solutions of almost radial nonlinear elliptic equations almost radial?, Commun. Part. Diff. Eq., (in press).
Z1 H. Zou, A local harnack inequality and classification of positive solutions of $\Delta u + u^p = 0$ in ${\mathbb{R}}^n$ , preprint.
Z2 H. Zou, Symmetry and local behavior of positive solutions of $\Delta u + u^p =0$ in ${\mathbb{R}}^n$ , preprint.