On Kirchhoff type problems involving critical and singular nonlinearities

Chun-Yu Lei¹ ; Chang-Mu Chu¹ ; Hong-Min Suo¹ ; Chun-Lei Tang²

¹ School of Sciences GuiZhou Minzu University 550025 Guiyang, China
² School of Mathematics and Statistics Southwest University 400715 Chongqing, China

Annales Polonici Mathematici, Tome 114 (2015) no. 3, pp. 269-291

Résumé

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 q2 $, $\lambda$ 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.

Détail

DOI : 10.4064/ap114-3-5

Keywords: paper interested multiple positive solutions kirchhoff type problem begin equation* cases int varomega nabla vardelta lambda frac q beta text varomega text partial varomega end equation* where varomega subset mathbb smooth bounded domain varomega lambda positive parameter beta satisfies inequalities obtain existence positive ground state solution multiple positive solutions via nehari manifold method