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 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 Not yet published.Expected publication date is January 6, 2014. 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. Table of Contents Introduction Main results Radial solutions in the power case Basic ingredients The analysis for the subcritical parameter The analysis for the critical parameter Illustration of our results Appendix A. Regular variation theory and related results Bibliography