eBook ISBN:  9780821890165 
Product Code:  MEMO/218/1027.E 
List Price:  $70.00 
MAA Member Price:  $63.00 
AMS Member Price:  $42.00 
eBook ISBN:  9780821890165 
Product Code:  MEMO/218/1027.E 
List Price:  $70.00 
MAA Member Price:  $63.00 
AMS Member Price:  $42.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 218; 2012; 105 ppMSC: Primary 35
The authors prove some refined asymptotic estimates for positive blowup solutions to \(\Delta u+\epsilon u=n(n2)u^{\frac{n+2}{n2}}\) on \(\Omega\), \(\partial_\nu u=0\) on \(\partial\Omega\), \(\Omega\) being a smooth bounded domain of \(\mathbb{R}^n\), \(n\geq 3\). In particular, they show that concentration can occur only on boundary points with nonpositive mean curvature when \(n=3\) or \(n\geq 7\). As a direct consequence, they prove the validity of the LinNi's conjecture in dimension \(n=3\) and \(n\geq 7\) for mean convex domains and with bounded energy. Recent examples by WangWeiYan show that the bound on the energy is a necessary condition.

Table of Contents

Chapters

Introduction

1. $L^\infty $bounded solutions

2. Smooth domains and extensions of solutions to elliptic equations

3. Exhaustion of the concentration points

4. A first upperestimate

5. A sharp upperestimate

6. Asymptotic estimates in $C^1\left (\Omega \right )$

7. Convergence to singular harmonic functions

8. Estimates of the interior blowup rates

9. Estimates of the boundary blowup rates

10. Proof of Theorems and

A. Construction and estimates on the Green’s function

B. Projection of the test functions


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Requests
The authors prove some refined asymptotic estimates for positive blowup solutions to \(\Delta u+\epsilon u=n(n2)u^{\frac{n+2}{n2}}\) on \(\Omega\), \(\partial_\nu u=0\) on \(\partial\Omega\), \(\Omega\) being a smooth bounded domain of \(\mathbb{R}^n\), \(n\geq 3\). In particular, they show that concentration can occur only on boundary points with nonpositive mean curvature when \(n=3\) or \(n\geq 7\). As a direct consequence, they prove the validity of the LinNi's conjecture in dimension \(n=3\) and \(n\geq 7\) for mean convex domains and with bounded energy. Recent examples by WangWeiYan show that the bound on the energy is a necessary condition.

Chapters

Introduction

1. $L^\infty $bounded solutions

2. Smooth domains and extensions of solutions to elliptic equations

3. Exhaustion of the concentration points

4. A first upperestimate

5. A sharp upperestimate

6. Asymptotic estimates in $C^1\left (\Omega \right )$

7. Convergence to singular harmonic functions

8. Estimates of the interior blowup rates

9. Estimates of the boundary blowup rates

10. Proof of Theorems and

A. Construction and estimates on the Green’s function

B. Projection of the test functions