A Supercritical Problem in Dimension Two

Abstract

Let $\Omega\subset\mathbb R^2$ be a smooth bounded domain containing the origin and invariant under reflection across the coordinate axes, and let $0<\lambda<\lambda_1(\Omega)$, where $\Lambda_1 (\Omega)$ is the first eigenvalue for $-\Delta$ on $\Omega$ under Dirichlet boundary conditions. For every fixed integer $k\ge1$ and all sufficiently small $\varepsilon>0$, we construct a positive solution of \[ -\Delta u=\lambda u e^{u^{2+\varepsilon}} \quad\hbox{in }\Omega, \qquad u=0\quad\hbox{on }\partial\Omega, \] 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.