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

Abstract

We study the concentration phenomenon of the Lane-Emden equation with vanishing potentials \[\begin{cases} -Δu_n=W_n(x)u_n^{p_n},\quad u_n>0,\quad\text{in}~Ω, u_n=0,\quad\text{on}~\partialΩ, \int_Ωp_n W_n(x)u_n^{p_n}dx\le C, \end{cases}\] where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^2$, $W_n(x)\ge 0$ are bounded functions with zeros in $\Omega$, and $p_n\to \infty$ as $n\to \infty$. A typical example is $W_n(x)=|x{|}^{2\alpha }$ with $0\in \Omega$, 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) \[p_n|x|^{2α}u_n(x)^{p_n-1+t}\to 8πe^{\frac{t}{2}}\sum_{i=1}^kδ_{a_i}+8π(1+α)e^{\frac{t}{2}}c^tδ_0, \quad t=0,1,2,\] for some $k\ge 0$, $a_i\in \Omega \setminus \{0\}$ and some $c\ge 1$. Moreover, for $\alpha \notin \mathbb{N}$, 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.