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Critical points of arbitrary energy for the Trudinger-Moser functional in planar domains

Andrea Malchiodi, Luca Martinazzi, Pierre-Damien Thizy

Abstract

Given a smoothly bounded non-contractible domain $Ω\subset \mathbb{R}^2$, we prove the existence of positive critical points of the Trudinger-Moser embedding for arbitrary Dirichlet energies. This is done via degree theory, sharp compactness estimates and a topological argument relying on the Poincaré-Hopf theorem.

Critical points of arbitrary energy for the Trudinger-Moser functional in planar domains

Abstract

Given a smoothly bounded non-contractible domain , we prove the existence of positive critical points of the Trudinger-Moser embedding for arbitrary Dirichlet energies. This is done via degree theory, sharp compactness estimates and a topological argument relying on the Poincaré-Hopf theorem.
Paper Structure (9 sections, 27 theorems, 171 equations, 3 figures)

This paper contains 9 sections, 27 theorems, 171 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorems \ref{['MainThm']} and \ref{['ThmDegree']}
  3. Degree formula for the vector field associated to \ref{['MeanFieldEquationh']} on a bounded domain $\Omega$
  4. Compactness of critical points of $\Phi_{N,h}$
  5. Bending the Robin function to apply the Poincaré-Hopf theorem
  6. Counting the new critical points
  7. Proof of Proposition \ref{['p:degree']} (completed)
  8. Properties of the Green's function
  9. The Euler characteristic of configuration spaces

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb{R}^2$ be a smoothly bounded non-contractible domain. Then, given any positive real number $\beta>0$, there exists a nonnegative function $u$, critical point of $F$ constrained to $\mathcal{M}_\beta\subset H^1_0$. In particular, $u$ is smooth and solves MainEq.

Figures (3)

  • Figure 1: Heuristic sketch of $\tilde{\mathcal{H}}$ near $\partial\Omega$
  • Figure 2: $(x_1,\dots, x_N)\in \Theta^*_{\delta,\delta',I}=((\Omega_\delta)^I\setminus D_I)\times ((\Sigma_{\delta'})^{N-I}\setminus D_{N-I})$
  • Figure :

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof : Proof of Proposition \ref{['PropMovPlane']} (completed)
  • Theorem 2.1
  • proof : Outline of the proof of Theorem \ref{['ThmBUp']}
  • Proposition 2.2
  • Remark 2.1
  • Theorem 2.2
  • Proposition 2.3: Chen-Lin ChenLin-Liouville
  • ...and 45 more