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The Lin-Ni's Problem for Mean Convex Domains
Olivier Druet, École Normale Supérieure de Lyon, France, Frédéric Robert, Université Henri Poincaré Nancy, Vandoeuvre-lès-Nancy, France, and Juncheng Wei, Chinese University of Hong Kong, Shatin, Hong Kong
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Memoirs of the American Mathematical Society
2012; 105 pp; softcover
Volume: 218
ISBN-10: 0-8218-6909-4
ISBN-13: 978-0-8218-6909-3
List Price: US$70 Individual Members: US$42
Institutional Members: US\$56
Order Code: MEMO/218/1027

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.

• Introduction
• $$L^\infty$$-bounded solutions
• Smooth domains and extensions of solutions to elliptic equations
• Exhaustion of the concentration points
• A first upper-estimate
• A sharp upper-estimate
• Asymptotic estimates in $$C^1\left(\Omega\right)$$
• Convergence to singular harmonic functions
• Estimates of the interior blow-up rates
• Estimates of the boundary blow-up rates
• Proof of Theorems 1 and 2
• Appendix A. Construction and estimates on the Green's function
• Appendix B. Projection of the test functions
• Bibliography