Energy Quantization of the Two Dimensional Lane-Emden Equation with Vanishing Potentials
Time: 2023-11-14
Published By: He Liu
Speaker(s): Houwang Li(Tsinghua University)
Time: 09:00-10:00 November 16, 2023
Venue: Room 78201, Jingchunyuan 78, BICMR
We study the concentration phenomenon of the Lane-Emden equation with vanishing potentials
\[
\left\{\begin{array}{l}
-\Delta u_n=W_n(x) u_n^{p_n}, \quad u_n>0, \quad \text { in } \Omega, \\
u_n=0, \quad \text { on } \partial \Omega \\
\int_{\Omega} p_n W_n(x) u_n^{p_n} \mathrm{~d} x \leq C,
\end{array}\right.\]
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^2, W_n(x) \geq 0$ are bounded functions with zeros in $\Omega$, and $p_n \rightarrow \infty$ as $n \rightarrow \infty$. 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 talk, we study the case $\alpha>0$ and prove a quantization property (suppose 0 is a concentration point)
\[
p_n|x|^{2 \alpha} u_n(x)^{p_n-1+t} \rightarrow 8 \pi e^{\frac{t}{2}} \sum_{i=1}^k \delta_{a_i}+8 \pi(1+\alpha) e^{\frac{t}{2}} c^t \delta_0, \quad t=0,1,2,
\]
for some $k \geq 0, a_i \in \Omega \backslash\{0\}$ and some $c \geq 1$. 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.
\[
\left\{\begin{array}{l}
-\Delta u_n=W_n(x) u_n^{p_n}, \quad u_n>0, \quad \text { in } \Omega, \\
u_n=0, \quad \text { on } \partial \Omega \\
\int_{\Omega} p_n W_n(x) u_n^{p_n} \mathrm{~d} x \leq C,
\end{array}\right.\]
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^2, W_n(x) \geq 0$ are bounded functions with zeros in $\Omega$, and $p_n \rightarrow \infty$ as $n \rightarrow \infty$. 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 talk, we study the case $\alpha>0$ and prove a quantization property (suppose 0 is a concentration point)
\[
p_n|x|^{2 \alpha} u_n(x)^{p_n-1+t} \rightarrow 8 \pi e^{\frac{t}{2}} \sum_{i=1}^k \delta_{a_i}+8 \pi(1+\alpha) e^{\frac{t}{2}} c^t \delta_0, \quad t=0,1,2,
\]
for some $k \geq 0, a_i \in \Omega \backslash\{0\}$ and some $c \geq 1$. 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.
