Energy quantization of the two dimensional Lane-Emden equation with vanishing potentials

We study the concentration phenomenon of the Lane-Emden equation with vanishing potentials $$\{\begin{array}{l}-\mathrm{\Delta }{u}_n=W_n(x)u_n^{p_n},u_n>0,\text{in }\mathrm{\Omega },u_n=0,\text{on }\mathrm{\partial }\mathrm{\Omega },{\int }_{\mathrm{\Omega }}{p}_n{W}_n(x)u_n^{p_n}dx\le C,\end{array}$$ where $\mathrm{\Omega }$ is a smooth bounded domain in ${\mathbb{R}}^2$, $W_n(x)\ge 0$ are bounded functions with zeros in $\mathrm{\Omega }$, and $p_n\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$. A typical example is $W_n(x)=\mathrm{\mid }x{\mathrm{\mid }}^{2\alpha }$ with $0\in \mathrm{\Omega }$, i.e. the equation turns to be the well-known Hénon equation. The asymptotic behavior for $\alpha =0$ has been well studied in the literature. While for $\alpha >0$, the problem becomes much more complicated since a singular Liouville equation appears as a limit problem. In this paper, we study the case $\alpha >0$ and prove a quantization property (suppose $0$ is a concentration point) $$p_n\mathrm{\mid }x{\mathrm{\mid }}^{2\alpha }{u}_n(x{)}^{p_n-1+t}\to 8\pi e^{\frac{t}{2}}\sum _{i=1}^k{\delta }_{a_i}+8\pi (1+\alpha )e^{\frac{t}{2}}{c}^t{\delta }_0,t=0,1,2,$$ for some $k\ge 0$, $a_i\in \mathrm{\Omega }\setminus \{0\}$ and some $c\ge 1$. Moreover, for $\alpha \in ̸\mathbb{N}$, we show that the blow up must be simple, i.e. $c=1$. As applications, we also obtain the complete asymptotic behavior of ground state solutions for the Hénon equation.