Abstract

We prove the existence of positive classical solutions for the p -Laplacian problem {−(r(t)ϕ(u′))′=−λuδ+f(t,u), t∈(0,1),u(0)=u(1)=0, where 0<δ<1 , ϕ(s)=|s|p−2s , p>1 , f:(0,1)×[0,∞)→R is a Carathéodory function satisfying lim supz→0+f(t,z)zp−1<λ1<lim infz→∞f(t,z)zp−1 uniformly for a.e. t ∈(0,1), where λ1 denotes the principal eigenvalue of −(r(t)ϕ(u′))′ with zero boundary conditions, and λ is a small nonnegative parameter.