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Positive solutions for critical elliptic systems in noncontractible domains
On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity
1.  Department of Mathematics, Henan Normal University, Xinxiang, 453007, China 
2.  Department of Mathematics, Dong Hua University, Shanghai, 200051, China 
$\Delta u=\lambda [\frac{1}{u^p}\frac{1}{u^q}]$ in $B$, $u=\kappa \in (0,(\frac{p1}{q1})^{1/(pq)} ]$ on $\partial B$, $0 < u < \kappa$
in $B$, where $p > q > 1$ and $B$ is the unit ball in $\mathbb R^N$ ($N \geq 2$). We show that there exists $\lambda_\star>0$ such that for $0<\lambda <\lambda_\star$, the maximal solution is the only positive radial solution. Furthermore, if $2 \leq N < 2+\frac{4}{p+1} (p+\sqrt{p^2+p})$, the branch of positive radial solutions must undergo infinitely many turning points as the maxima of the radial solutions on the branch go to 0. The key ingredient is the use of a monotonicity formula.
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