Critical points of the Moser-Trudinger functional

TL;DR

This paper investigates the blow-up behavior of solutions to a heat flow related to the Moser-Trudinger functional on 2D domains, establishing conditions for critical points and their topological dependence.

Contribution

It characterizes the blow-up limits of solutions, links topological properties of the domain to the existence of critical points, and demonstrates the naturality of topological assumptions.

Findings

Blow-up solutions converge weakly to zero with energy quantized in multiples of 4π.

Existence of positive critical points depends on the domain's topology and the parameter Λ.

No positive critical points exist for large Λ in the unit ball, confirming the topological condition's necessity.

Abstract

On a smooth bounded 2-dimensional domain $\Omega$ we study the heat flow $u_t=\Delta u +\lambda (t)ue^{u^2}$ ($\lambda(t)$ is such that $d/dt ||u(t,\cdot)||_{H^1_0}=0$) introduced by T. Lamm, F. Robert and M. Struwe to investigate the Moser-Trudinger functional $E(v)=\int_{\Omega} (e^{v^2}-1)dx, v\in H^1_0(\Omega).$ We prove that if $u$ blows-up as $t\to\infty$ and if $E(u(t,\cdot))$ remains bounded, then for a sequence $t_k\to\infty$ we have $u(t_k,\cdot)\rightharpoonup 0$ in $H^1_0$ and $\|u(t_k,\cdot)\|_{H^1_0}^2\to 4\pi L$ for an integer $L\ge 1$. We couple these results with a topological technique to prove that if $\Omega$ is not contractible, then for every $0<\Lambda\in \mathbb{R} \setminus 4 \pi \mathbb{N}$ the functional $E$ constrained to $M_\Lambda=\{v\in H^1_0(\Omega):||v||_{H^1_0}^2=\Lambda \}$ has a positive critical point. We prove that when $\Omega$ is the unit ball and $\Lambda$ is large enough, then $E|_{M_\Lambda}$ has no positive critical points, hence showing that the topological assumption on $\Omega$ is natural.