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

This work is devoted to the study of blow-up phenomena for solutions of $$-\mathrm{\Delta }u=\lambda V(x)\mathrm{\mid }x{\mathrm{\mid }}^{2N}{e}^v,in{B}_1,v=0on\mathrm{\partial }{B}_1\mathrm{.}$$ 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 $\lambda >0$ 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 $\lambda \to 0^{+}$. 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.