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A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials
Florica C. Cîrstea, University of Sydney, Australia
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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
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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
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