Existence of a positive solution for one-dimensional singular p-Laplacian problems and its parameter dependence

We consider one-dimensional singular p-Laplacian problems of the form { λ ( φ p ( u ′ ) ) ′ + ω ( t ) f ( t , u ) = 0 , t ∈ ( 0 , 1 ) , a u ( 0 ) − b u ′ ( 0 ) = ∫ 0 1 g ( t ) u ( t ) d t , u ′ ( 1 ) = 0 , where λ is a positive parameter, φ p ( s ) = | s | p − 2 s , p > 1 , ( φ p ) − 1 = φ q , 1 p + 1 q = 1 , and ω may be singular at t = 0 and/or t = 1 . Using fixed-point techniques combined with the partially ordered structure of Banach space, we establish some new and more general existence, multiplicity, and nonexistence results. We also study the dependence of the positive solution u λ ( t ) on the parameter λ, that is, lim λ → + ∞ ‖ u λ ‖ = + ∞ or lim λ → + ∞ ‖ u λ ‖ = 0 . An example illustrates our main results.