Existence and multiplicity of solutions for a degenerate nonlocal elliptic differential equation

Using variational arguments, we study the existence and multiplicity of solutions for the degenerate nonlocal differential equation $$displaylines{ - MBig(int_Omega |x|^{-ap}|abla u|^p,dxBig)operatorname{div} Big(|x|^{-ap}|abla u|^{p-2}abla uBig) = |x|^{-p(a+1)+c} f(x,u) quad hbox{in } Omega,cr u = 0 quad hbox{on } partialOmega, }$$ where $Omega subset mathbb{R}^N$ ($N geq 3$) and the function M may be zero at zero.