Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents (Q401381)

For the following Dirichlet boundary problem of Kirchhoff type equation NEWLINE\[NEWLINE -(a + b\int_\Omega|\nabla u|^2 dx) \Delta u= u^5 + \lambda u^{-\gamma} \text{ in } \Omega\subset \mathbb R^3, \;u|_{\partial\Omega}= 0, \text{ and } a>0, \;b\geq 0, \;\gamma \in (0,1), NEWLINE\]NEWLINE the authors prove that there is \(\lambda_* >0\) such that, for any \(\lambda \in (0,\lambda_*)\), the above problem admits two distinct positive solutions, by using variational and perturbation methods.