Abstract

We discuss the existence of positive solutions of the problem - (q (t) f (u' (t)))' = f (t, u (t), u' (t)) for t ? (0, 1) and u (0) = u (1) = 0, where the nonlinearity f satisfies a superlinearity condition at 0 and a local superlinearity condition at + 8. This general quasilinear differential operator involves a weight q and a main differentiable part f which is not necessarily a power. Due to the superlinearity of f and its dependence on the derivative, a condition of the Bernstein-Nagumo type is assumed, also involving the differential operator. Our main result is the proof of a priori bounds for the eventual solutions. The presence of the derivative in the right-hand side of the equation requires a priori bounds not only on the solutions themselves, but also on their derivatives, which brings additional difficulties. As an application, we consider a quasilinear Dirichlet problem in an annulus{(- div (A (| ? u |) ? u) = f (| x |, u, | ? u |) in r1 < | x | < r2,; u (x) = 0 on | x | = R1 and | x | = R2 .). © 2009 Elsevier Ltd. All rights reserved.