Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Localization of bubbling for high order nonlinear equations

Frédéric Robert

Abstract

We analyze the asymptotic pointwise behavior of families of solutions to the high-order critical equation $$P_αu_α=Δ_g^k u_α+\hbox{lot}=|u_α|^{2^\star-2-ε_α} u_α\hbox{ in }M$$ that behave like $$u_α=u_0+B_α+o(1)\hbox{ in }H_k^2(M)$$ where $B=(B_α)_α$ is a Bubble, also called a Peak. We give obstructions for such a concentration to occur: depending on the dimension, they involve the mass of the associated Green's function or the difference between $P_α$ and the conformally invariant GJMS operator. The bulk of this analysis is the proof of the pointwise control \begin{equation*} |u_α(x)|\leq C\Vert u_0\Vert_\infty^{(2^\star-1)^2}+C\left(\frac{μ_α^{2}}{μ_α^{2 }+d_g(x,x_α)^{2 }}\right)^{\frac{n-2k}{2}}\hbox{ for all }x\in M\hbox{ and }α\in\mathbb{N}, \end{equation*} where $|u_α(x_α)|=\max_M|u_α|\to +\infty$ and $μ_α:=|u_α(x_α)|^{-\frac{2}{n-2k}}$. The key to obtain this estimate is a sharp control of the Green's function for elliptic operators involving a Hardy potential.

Localization of bubbling for high order nonlinear equations

Abstract

We analyze the asymptotic pointwise behavior of families of solutions to the high-order critical equation that behave like where is a Bubble, also called a Peak. We give obstructions for such a concentration to occur: depending on the dimension, they involve the mass of the associated Green's function or the difference between and the conformally invariant GJMS operator. The bulk of this analysis is the proof of the pointwise control \begin{equation*} |u_α(x)|\leq C\Vert u_0\Vert_\infty^{(2^\star-1)^2}+C\left(\frac{μ_α^{2}}{μ_α^{2 }+d_g(x,x_α)^{2 }}\right)^{\frac{n-2k}{2}}\hbox{ for all }x\in M\hbox{ and }α\in\mathbb{N}, \end{equation*} where and . The key to obtain this estimate is a sharp control of the Green's function for elliptic operators involving a Hardy potential.
Paper Structure (31 sections, 33 theorems, 409 equations)

This paper contains 31 sections, 33 theorems, 409 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. The notion of bubble and the product of the Weyl tensor and a bubble
  3. The product $\hbox{Weyl}_g\otimes B$
  4. More general results
  5. Generalization of Theorem \ref{['th:estim-co:intro']}
  6. Generalizations of Theorems \ref{['th:teaser:2']} and \ref{['th:teaser:3']}
  7. Other consequences
  8. Preliminary material: Sobolev and Hardy inequalities
  9. Preliminary analysis: adapted chart and convergence up to rescaling
  10. Strong pointwise control: proof of Theorem \ref{['th:estim-co:intro']}
  11. Green's function with Hardy potential: construction and first estimates
  12. The Hardy-type potential
  13. Asymptotics for the Green's function close to the singularity
  14. Asymptotics for the Green's function far from the singularity
  15. Proof of Theorem \ref{['th:Green:pointwise:BIS']}
  16. ...and 16 more sections

Key Result

Theorem 1.1

[Standard bubble] Let $(M,g)$ be a compact Riemannian manifold of dimension $n$ without boundary and let $k\in\mathbb{N}$ be such that $2\leq 2k<n$. Let $u_0\in C^{2k}(M)$ be a function (possibly sign-changing), $x_0\in M$ be a point, and $(P_\alpha)_{\alpha>0}\to P_0$ be of type (SCC). Assume that where $A_{g}^{(k-1)}$ is the coefficient of the term of order $2k-2$ in the GJMS operator def:gjms.

Theorems & Definitions (50)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2: Exponential chart
  • Definition 2.3
  • Proposition 2.1
  • Definition 2.4
  • Proposition 2.2
  • ...and 40 more