On Kirchhoff type problems involving critical and singular nonlinearities
Tom 114 / 2015
Annales Polonici Mathematici 114 (2015), 269-291
MSC: Primary 35R09; Secondary 35A15, 35B09.
DOI: 10.4064/ap114-3-5
Streszczenie
In this paper, we are interested in multiple positive solutions for the Kirchhoff type problem \begin{equation*} \cases{ -(a+b\int_\varOmega|\nabla u|^2\,dx) \varDelta u=u^{5}+\lambda\frac{u^{q-1}}{|x|^\beta} & \text{in $\varOmega$,} \cr u=0 &\text{on $\partial\varOmega$,} \cr} \end{equation*} where $\varOmega\subset \mathbb{R}^{3}$ is a smooth bounded domain, $0\in\varOmega$, $1< q<2 $, $\lambda$ is a positive parameter and $\beta$ satisfies some inequalities. We obtain the existence of a positive ground state solution and multiple positive solutions via the Nehari manifold method.