Existence of large solutions for a class of quasilinear elliptic equations

We consider the quasilinear elliptic equationdiv(|∇u|p-2∇u)=a(x)f(u),u⩾0inΩ,where p>1,a(x)⩾0,a(x)≢0,a(x)∈C(Ω¯), and f is continuous and non-decreasing on [0,+∞), satisfies f(0)=0,f(s)>0fors>0 and the Keller-Osserman condition∫1+∞[F(s)]-1/pds=+∞,F(s)=∫0sf(t)dt.We establish conditions on the function a that are necessary and sufficient for the existence of positive solutions, bounded and unbounded, of the given equation.