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

Abstract

This study examines nonnegative solutions to the problem \begin{equation*}\left\{\arraycolsep=1.5pt \begin {array}{lll} \Delta u=\displaystyle\frac{\lambda|x|^{\alpha}}{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 $\lam >0,$ $\alp>-2$, and $p>0$ are constants. The possible asymptotic behaviors of $u(x)$ at $|x|=0$ and $|x|=\infty$ are classified according to $(\alpha,p)$. In particular, the results show that for some $(\alpha,p)$, $u(x)$ exhibits only ``isotropic" behavior at $|x|=0$ and $|x|=\infty$. However, in other cases, $u(x)$ may exhibit the "anisotropic" behavior at $|x|=0$ or $|x|=\infty$. Furthermore, the relation between the limit at $|x|=0$ and the limit at $|x|=\infty$ for a global solution is investigated.