July  2008, 7(4): 765-786. doi: 10.3934/cpaa.2008.7.765

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

1. 

Department of Mathematics, Henan Normal University, Xinxiang, 453002, China

2. 

Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

Received  April 2007 Revised  January 2008 Published  April 2008

We consider the following problem

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

Citation: Zongming Guo, Juncheng Wei. Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity. Communications on Pure & Applied Analysis, 2008, 7 (4) : 765-786. doi: 10.3934/cpaa.2008.7.765
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