Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Blow-up of solutions of critical elliptic equations in three dimensions

Rupert L. Frank, Tobias König, Hynek Kovařík

Abstract

We describe the asymptotic behavior of positive solutions $u_ε$ of the equation $-Δu + au = 3\,u^{5-ε}$ in $Ω\subset\mathbb{R}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon and the functions $u_ε$ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brézis and Peletier (1989). Similar results are also obtained for solutions of the equation $-Δu + (a+εV) u = 3\,u^5$ in $Ω$.

Blow-up of solutions of critical elliptic equations in three dimensions

Abstract

We describe the asymptotic behavior of positive solutions of the equation in with a homogeneous Dirichlet boundary condition. The function is assumed to be critical in the sense of Hebey and Vaugon and the functions are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brézis and Peletier (1989). Similar results are also obtained for solutions of the equation in .

Paper Structure

This paper contains 37 sections, 49 theorems, 382 equations.

Table of Contents

  1. Introduction and main results
  2. Additive case: A first expansion
  3. A qualitative initial expansion
  4. Coercivity
  5. The bound on $\|\nabla w\|_2$
  6. Excluding boundary concentration
  7. Proof of Proposition \ref{['prop first expansion']}
  8. Additive case: Refining the expansion
  9. Bounds on $s$
  10. The bound on $\|\nabla r\|_2$
  11. Expanding $\alpha^4$
  12. Expanding $\phi_a(x)$
  13. Bounding $\nabla \phi_a(x)$
  14. Proof of Theorems \ref{['thm expansion']} and \ref{['thm BP']}
  15. The behavior of $\phi_a$ near $x_0$
  16. ...and 22 more sections

Key Result

Theorem \oldthetheorem

Let $(u_\varepsilon)$ be a family of solutions to BP-problem satisfying eq:sobmineps. Then there are sequences $(x_\varepsilon)\subset\Omega$, $(\lambda_\varepsilon)\subset(0,\infty)$, $(\alpha_\varepsilon)\subset\mathbb{R}$ and $(r_\varepsilon) \subset T_{x_\varepsilon, \lambda_\varepsilon}^\bot$ s and a point $x_0 \in \Omega$ with $\nabla\phi_a(x_0)=0$ such that, along a subsequence, Moreover,

Theorems & Definitions (86)

  • Theorem \oldthetheorem: Asymptotic expansion of $u_\varepsilon$
  • Theorem \oldthetheorem: Brézis-Peletier conjecture
  • Theorem \oldthetheorem: Asymptotic expansion of $u_\varepsilon$
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • ...and 76 more