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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

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
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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

  • 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
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