**Memoirs of the American Mathematical Society**

2012;
105 pp;
Softcover

MSC: Primary 35;

Print ISBN: 978-0-8218-6909-3

Product Code: MEMO/218/1027

List Price: $70.00

AMS Member Price: $42.00

MAA Member Price: $63.00

**Electronic ISBN: 978-0-8218-9016-5
Product Code: MEMO/218/1027.E**

List Price: $70.00

AMS Member Price: $42.00

MAA Member Price: $63.00

# The Lin-Ni’s Problem for Mean Convex Domains

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*Olivier Druet; Frédéric Robert; Juncheng Wei*

The authors prove some refined asymptotic estimates for positive blow-up solutions to \(\Delta u+\epsilon u=n(n-2)u^{\frac{n+2}{n-2}}\) 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 Lin-Ni's conjecture in dimension \(n=3\) and \(n\geq 7\) for mean convex domains and with bounded energy. Recent examples by Wang-Wei-Yan show that the bound on the energy is a necessary condition.

#### Table of Contents

# Table of Contents

## The Lin-Ni's Problem for Mean Convex Domains

- Introduction 18 free
- Chapter 1. L-bounded solutions 512 free
- Chapter 2. Smooth domains and extensions of solutions to elliptic equations 714
- Chapter 3. Exhaustion of the concentration points 1118
- Chapter 4. A first upper-estimate 1926
- Chapter 5. A sharp upper-estimate 2734
- Chapter 6. Asymptotic estimates in C1() 4350
- Chapter 7. Convergence to singular harmonic functions 4552
- Chapter 8. Estimates of the interior blow-up rates 5764
- Chapter 9. Estimates of the boundary blow-up rates 6976
- Chapter 10. Proof of Theorems 1 and 2 8188
- Appendix A. Construction and estimates on the Green's function 8390
- Appendix B. Projection of the test functions 97104
- Bibliography 103110