eBook ISBN:  9781470414290 
Product Code:  MEMO/227/1068.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 
eBook ISBN:  9781470414290 
Product Code:  MEMO/227/1068.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 227; 2014; 85 ppMSC: 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 \(\infty<\lambda\leq (N2)^2/4\) 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


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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 \(\infty<\lambda\leq (N2)^2/4\) 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