Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity

$-\Delta u=\frac{\lambda}{(1-u)^2}$ in $\Omega$, $u=0$ on $\partial \Omega$, $0 < u < 1$ in $\Omega$

where $\Omega$ is a rather symmetric domain in $\mathbb R^2$. We prove that there exists a $\lambda_\star>0$ such that for $\lambda \in (0, \lambda_\star)$ the minimal solution is unique. Then we analyze the asymptotic behavior of touch-down solutions, i.e., solutions with max$_\Omega u_i (0) \to 1$. We show that after a rescaling, the solution will be asymptotically symmetric. As a consequence, we show that the branch of positive solutions must undergo infinitely many bifurcations as the maximums of the solutions on the branch go to 1 (possibly only changes of direction). This gives a positive answer to some open problems in [12]. Our result is new even in the radially symmetric case. Central to our analysis is the monotonicity formula, one-dimensional Sobloev inequality, and classification of solutions to a supercritical problem

$ \Delta U=\frac{1}{U^2}\quad$ in $\mathbb R^2, U(0)=1, U(z) \geq 1.$