# A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials

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*Florica C. Cîrstea*

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

# Table of Contents

## A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials

- Chapter 1. Introduction 18 free
- Chapter 2. Main results 1522
- Chapter 3. Radial solutions in the power case 2330
- Chapter 4. Basic ingredients 2936
- Chapter 5. The analysis for the subcritical parameter 3744
- Chapter 6. The analysis for the critical parameter 6168
- Chapter 7. Illustration of our results 7582
- Appendix A. Regular variation theory and related results 7986
- Bibliography 8390