Quasilinear equations with dependence on the gradient

Abstract

We discuss the existence of positive solutions of the problem − ( q ( t ) φ ( 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 + ∞ . This general quasilinear differential operator involves a weight q and a main differentiable part φ 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 r 1 < | x | < r 2 , u ( x ) = 0 on | x | = R 1 and | x | = R 2 .