New Titles  |  FAQ  |  Keep Informed  |  Review Cart  |  Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education

A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials
Florica C. Cîrstea, University of Sydney, Australia
 SEARCH THIS BOOK:
Memoirs of the American Mathematical Society
2014; 85 pp; softcover
Volume: 227
ISBN-10: 0-8218-9022-0
ISBN-13: 978-0-8218-9022-6
List Price: US$71 Individual Members: US$42.60
Institutional Members: US\$56.80
Order Code: MEMO/227/1068

In this paper, the author considers semilinear elliptic equations of the form $$-\Delta u- \frac{\lambda}{|x|^2}u +b(x)\,h(u)=0$$ in $$\Omega\setminus\{0\}$$, where $$\lambda$$ is a parameter with $$-\infty<\lambda\leq (N-2)^2/4$$ and $$\Omega$$ is an open subset in $$\mathbb{R}^N$$ with $$N\geq 3$$ such that $$0\in \Omega$$. Here, $$b(x)$$ is a positive continuous function on $$\overline \Omega\setminus\{0\}$$ which behaves near the origin as a regularly varying function at zero with index $$\theta$$ greater than $$-2$$. The nonlinearity $$h$$ is assumed continuous on $$\mathbb{R}$$ and positive on $$(0,\infty)$$ with $$h(0)=0$$ such that $$h(t)/t$$ is bounded for small $$t>0$$. The author completely classifies the behaviour near zero of all positive solutions of equation (0.1) when $$h$$ is regularly varying at $$\infty$$ with index $$q$$ greater than $$1$$ (that is, $$\lim_{t\to \infty} h(\xi t)/h(t)=\xi^q$$ for every $$\xi>0$$). In particular, the author's results apply to equation (0.1) with $$h(t)=t^q (\log t)^{\alpha_1}$$ as $$t\to \infty$$ and $$b(x)=|x|^\theta (-\log |x|)^{\alpha_2}$$ as $$|x|\to 0$$, where $$\alpha_1$$ and $$\alpha_2$$ are any real numbers.