The Lin-Ni’s problem for mean convex domains

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...

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Bibliographic Details
Main Authors: Druet, Olivier, Robert, Frédéric, Wei, Juncheng
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2011
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Blowing up (Algebraic geometry)
Convex domains
Differential equations, Elliptic
Neumann problem
Neumann elliptic problem
critical exponent
blow-up
ISBN:9780821869093, 0821869094
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Abstract 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.
AbstractList 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.
We prove some refined asymptotic estimates for positive blow-up solutions to
Author Wei, Juncheng
Robert, Frédéric
Druet, Olivier
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Copyright Copyright 2011 American Mathematical Society
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Keywords Neumann elliptic problem
critical exponent
blow-up
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Notes "July 2012, volume 218, number 1027 (end of volume)"--T.p
Includes bibliography (p. 103-105)
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Snippet We prove some refined asymptotic estimates for positive blow-up solutions to
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 $\Omega$,...
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SubjectTerms Blowing up (Algebraic geometry)
Convex domains
Differential equations, Elliptic
Neumann problem
TableOfContents Introduction -- <inline-formula content-type="math/mathml"> L ∞ − L^\infty - </inline-formula>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 <inline-formula content-type="math/mathml"> C 1 ( Ω ) C^1\left (\Omega \right ) </inline-formula> -- Convergence to singular harmonic functions -- Estimates of the interior blow-up rates -- Estimates of the boundary blow-up rates -- Proof of Theorems and -- Construction and estimates on the Green’s function -- Projection of the test functions
Intro -- Contents -- Abstract -- Introduction -- Chapter 1. L-bounded solutions -- Chapter 2. Smooth domains and extensions of solutions to elliptic equations -- Chapter 3. Exhaustion of the concentration points -- Chapter 4. A first upper-estimate -- Chapter 5. A sharp upper-estimate -- Chapter 6. Asymptotic estimates in C1() -- Chapter 7. Convergence to singular harmonic functions -- 1. Convergence at general scale -- 2. Convergence at appropriate scale -- Chapter 8. Estimates of the interior blow-up rates -- Chapter 9. Estimates of the boundary blow-up rates -- Chapter 10. Proof of Theorems 1 and 2 -- Appendix A. Construction and estimates on the Green's function -- Appendix B. Projection of the test functions -- Bibliography
Title The Lin-Ni’s problem for mean convex domains
URI https://www.ams.org/memo/1027/
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Volume 218
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