Study on the behaviors of rupture solutions for a class of elliptic MEMS equations in R2

Abstract

This study examines nonnegative solutions to the problem$\{\begin{array}{cc}\Delta u=\frac{\lambda |x{|}^{\alpha }}{u^p}& \text{ in}{R}^2\setminus \left\{0\right\},\\ u\left(0\right)=0\text{and}u>0& \text{ in}{R}^2\setminus \left\{0\right\},\end{array}$ where $\lambda >0$, $\alpha >-2$, and $p>0$ are constants. The possible asymptotic behaviors of $u\left(x\right)$ at $\left|x\right|=0$ and $\left|x\right|=\infty$ are classified according to $(\alpha ,p)$. In particular, the results show that for some $(\alpha ,p)$, $u\left(x\right)$ exhibits only “isotropic” behavior at $\left|x\right|=0$ and $\left|x\right|=\infty$. However, in other cases, $u\left(x\right)$ may exhibit the “anisotropic” behavior at $\left|x\right|=0$ or $\left|x\right|=\infty$. Furthermore, the relation between the limit at $\left|x\right|=0$ and the limit at $\left|x\right|=\infty$ for a global solution is investigated.