Higher-order criteria and sharpness for blow-up at quantized singularities in Liouville equations

Abstract

This work is devoted to the study of blow-up phenomena for solutions of \[-Δu=λV(x)|x|^{2N}e^v,\quad \mbox{in}\quad B_1,\quad v=0\quad \mbox{on}\quad \partial B_1. \] where $B_1$ denotes the unit ball in $\mathbb R^2$ centered at the origin, $V(x)$ is a positive smooth function, $N\in \mathbb N$ and $λ>0$ is a small parameter. The main goal is to determine conditions on the potential $V$ guaranteeing the existence of families of solutions that blow up at the origin as $λ\to 0^+$. Moreover, we establish higher order criteria that distinguish between simple and non-simple blow-up behavior at $x=0$. In particular, for $N-1\in 2^{\mathbb N}$, we construct a non-simple blow-up sequence $v_k$, corresponding to coefficient function $V_k$ whose derivatives vanish up to the maximal order allowed by the upper bound of Wei-Zhang, thereby proving that this bound is sharp.