Sign-changing blow-up for the Moser-Trudinger equation (Q2056412)

In this interesting paper, the authors consider a domain \(\Omega\Subset \mathbb{R}^2\) satisfying certain symmetry conditions. They prove that for any \(k\in\mathbb{N}\setminus\{0\}\) and \(\beta>4\pi k\), one can construct blowing-up solutions \((u_{\varepsilon})\subset H^1_0(\Omega)\) to the Moser-Trudinger equation such that as \(\varepsilon\rightarrow 0\), the following three statements hold: \begin{itemize} \item \(||\nabla u_{\varepsilon}||_{L^2}^2\rightarrow \beta\); \item \(u_{\varepsilon}\rightharpoonup u_0\) in \(H_0^1\), where \(u_0\) is a sign-changing solution of the Moser-Trudinger equation; \item \(u_{\varepsilon}\) develops \(k\) positive spherical bubbles, all concentrating at \(0\in \Omega\). \end{itemize} The lack of quantization, non-zero weak limit and bubble clustering stand in sharp contrast to the case \(u_{\varepsilon}>0\) studied in [\textit{O. Druet} and the second author, J. Eur. Math. Soc. (JEMS) 22, No. 12, 4025--4096 (2020; Zbl 1458.35167)], where blow-up can happen only when \(\beta\in 4\pi\mathbb{N}\), the weak limit vanishes, and the bubbles blow up at distinct point. The main result is proved by constructing a sign-changing weak limit, and using a Lyapunov-Schmidt procedure to glue to it an arbitrary number \(k\in\mathbb{N}\setminus\{0\}\) of bubbles, all concentrating at the origin.