Sign-changing blow-up for the Moser-Trudinger equation
Résumé
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 $\left(u_\varepsilon\right)\subset H^1_0\left(\Omega\right)$ to the Moser--Trudinger equation such that as $\varepsilon\downarrow 0$, we have $\left\|\nabla u_\varepsilon\right\|_{L^2}^2\to \beta$, $u_\varepsilon \rightharpoonup u_0$ 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 [8].
Origine : Fichiers produits par l'(les) auteur(s)