Existence of positive solutions for some problems with nonlinear diffusion

In this paper we study the existence of positive solutions for problems of the type − Δ p u ( x ) a m p ; = u ( x ) q − 1 h ( x , u ( x ) ) , a m p ; a m p ; x ∈ Ω , u ( x ) a m p ; = 0 , a m p ; a m p ; x ∈ ∂ Ω , {equation*} {aligned} -Δ _pu(x) &=u(x)q-1h(x,u(x)), && x∈ Ω , \\ u(x)&=0, && x∈ ∂ Ω , {aligned} {equation*} where Δ p Δ _p is the p p -Laplace operator and p , q > 1 p,q>1 . If p = 2 p=2 , such problems arise in population dynamics. Making use of different methods (sub- and super-solutions and a variational approach), we treat the cases