Abstract

We prove the existence of positive classical solutions for the pLaplacian problem −(r(t)φ(u 0 ))0 = f(t, u), t ∈ (0, 1), au(0) − bφ−1 (r(0))u 0 (0) = 0, cu(1) + dφ−1 (r(1))u 0 (1) = 0, where φ(s) = |s| p−2s, p > 1, f : (0, 1)×[0, ∞) → R is a Carath´eodory function satisfying lim sup z→0+ f(t, z) z p−1 < λ1 < lim inf z→∞ f(t, z) z p−1 uniformly for a.e. t ∈ (0, 1), where λ1 denotes the principal eigenvalue of −(r(t)φ(u 0 ))0 with Sturm-Liouville boundary conditions. Our result extends a previous work by Man´asevich, Njoku, and Zanolin to the Sturm-Liouville boundary conditions with more general operator.