A Supercritical Problem in Dimension Two

Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be a smooth bounded domain containing the origin and invariant under reflection across the coordinate axes, and let $0<\lambda <{\lambda }_1(\mathrm{\Omega })$, where ${\mathrm{\Lambda }}_1(\mathrm{\Omega })$ is the first eigenvalue for $-\mathrm{\Delta }$ on $\mathrm{\Omega }$ under Dirichlet boundary conditions. For every fixed integer $k\ge 1$ and all sufficiently small $\epsilon >0$, we construct a positive solution of $$-\mathrm{\Delta }u=\lambda u{e}^{u^{2+\epsilon }}\text{in }\mathrm{\Omega },u=0\text{on }\mathrm{\partial }\mathrm{\Omega },$$ which blows up at the origin as a tower of $k$ Liouville bubbles. The concentration scales are strongly separated. The proof is based on a Lyapunov–Schmidt reduction adapted to these different scales.