Sign-changing blow-up for the Moser-Trudinger equation

TL;DR

This paper constructs sign-changing solutions to the Moser-Trudinger equation on symmetric domains that blow up with multiple bubbles, exhibit non-quantized energy, and have a non-zero weak limit, contrasting with positive solutions.

Contribution

It introduces the first construction of sign-changing blow-up solutions with bubble clustering and non-quantized energy for the Moser-Trudinger equation.

Findings

Solutions develop multiple positive bubbles concentrating at a point.

Weak limits of solutions are sign-changing, non-trivial functions.

The blow-up behavior differs from positive solutions, showing lack of quantization.

Abstract

Given a sufficiently symmetric domain $\Omega\Subset\mathbb{R}^2$, for any $k\in \mathbb{N}\setminus \{0\}$ and $\beta>4\pi k$ we construct blowing-up solutions $(u_\varepsilon)\subset H^1_0(\Omega)$ to the Moser-Trudinger equation such that as $\varepsilon\downarrow 0$, we have $\|\nabla u_\varepsilon\|_{L^2}^2\to \beta$, $u_\varepsilon \rightharpoonup u_0$ in $H^1_0$ where $u_0$ is a sign-changing solution of the Moser-Trudinger equation and $u_\varepsilon$ develops $k$ positive spherical bubbles, all concentrating at $0\in \Omega$. These $3$ features (lack of quantization, non-zero weak limit and bubble clustering) stand in sharp contrast to the positive case ($u_\varepsilon>0$) studied by the second author and Druet (J. Eur. Math. Soc. (JEMS) 22 (2020)).