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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, Sydney, Australia
A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials
eBook ISBN:  978-1-4704-1429-0
Product Code:  MEMO/227/1068.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials
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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, Sydney, Australia
eBook ISBN:  978-1-4704-1429-0
Product Code:  MEMO/227/1068.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2272014; 85 pp
    MSC: Primary 35;

    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
     
     
    • Chapters
    • 1. Introduction
    • 2. Main results
    • 3. Radial solutions in the power case
    • 4. Basic ingredients
    • 5. The analysis for the subcritical parameter
    • 6. The analysis for the critical parameter
    • 7. Illustration of our results
    • A. Regular variation theory and related results
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2272014; 85 pp
MSC: Primary 35;

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.

  • Chapters
  • 1. Introduction
  • 2. Main results
  • 3. Radial solutions in the power case
  • 4. Basic ingredients
  • 5. The analysis for the subcritical parameter
  • 6. The analysis for the critical parameter
  • 7. Illustration of our results
  • A. Regular variation theory and related results
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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