Study on the behaviors of rupture solutions for a class of elliptic MEMS equations in ${\mathbb{R}}^2$

This study examines nonnegative solutions to the problem \begin{equation*}\left{\arraycolsep=1.5pt \begin {array}{lll} Δu=\displaystyle\frac{λ|x|^α}{u^p} \ \ &\hbox{ in} ,\ \R ^2\setminus {0},\[2mm] u(0)=0 \ $\text{and}$\ u> 0 \ \ &\hbox{ in} ,\ \R ^2\setminus {0},\ \end{array}\right. \label{eqn} \end{equation*} where $\text{lam}>0,$ $\text{alp}>-2$, and $p>0$ are constants. The possible asymptotic behaviors of $u(x)$ at $\mathrm{\mid }x\mathrm{\mid }=0$ and $\mathrm{\mid }x\mathrm{\mid }=\mathrm{\infty }$ are classified according to $(\alpha ,p)$. In particular, the results show that for some $(\alpha ,p)$, $u(x)$ exhibits only ``isotropic" behavior at $\mathrm{\mid }x\mathrm{\mid }=0$ and $\mathrm{\mid }x\mathrm{\mid }=\mathrm{\infty }$. However, in other cases, $u(x)$ may exhibit the “anisotropic” behavior at $\mathrm{\mid }x\mathrm{\mid }=0$ or $\mathrm{\mid }x\mathrm{\mid }=\mathrm{\infty }$. Furthermore, the relation between the limit at $\mathrm{\mid }x\mathrm{\mid }=0$ and the limit at $\mathrm{\mid }x\mathrm{\mid }=\mathrm{\infty }$ for a global solution is investigated.