Nonlinear Analysis 71 (2009) 4862–4868 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Quasilinear equations with dependence on the gradient Djairo G. De Figueiredo a , Justino Sánchez b,c,∗ , Pedro Ubilla c a IMECC, UNICAMP, Caixa Postal 6065, 13081-970, Brazil b Departamento de Matemáticas, Universidad de la Serena, Casilla 559-554, La Serena, Chile c Departamento de Matemática y C.C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile article info Article history: Received 27 October 2008 Accepted 20 March 2009 MSC: 34B15 34B18 35J60 35J65 Keywords: Positive radial solutions Fixed points Annular domains Bernstein–Nagumo condition 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 superlin- earity 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 nec- essarily 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 opera- tor. 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 . © 2009 Elsevier Ltd. All rights reserved. 1. Introduction In the study of quasilinear elliptic equations in annular domains in R N the question on the existence of radial solutions appears quite naturally. The PDE problem is then transformed into an equivalent ordinary differential equation. In Section 4 we discuss in detail a Dirichlet problem for a nonlinear elliptic equation in an annular domain in R N . In this way we see that the nonlinear ordinary differential equations studied in this paper have their motivation in the question of obtaining radial solutions for some nonlinear partial differential equations. Our main concerns here are the consideration of nonlinearities with dependence on the gradient, and nonlinear differential operators of a very general type. As a consequence of the presence of the derivative in the nonlinearity, this type of problem cannot be treated by Variational Methods. So our approach is the use of Topological Degree, via a theorem of Krasnosel’skii for mappings defined in cones. For that matter, one of the main difficulties is obtaining a priori bounds on the derivatives of the solutions. Let us consider the Dirichlet problem for the following nonlinear ordinary differential equation:  −(q(t )ϕ(u ′ (t ))) ′ = f (t , u(t ), u ′ (t )) for t ∈ (0, 1), u(0) = u(1) = 0. (1.1) As we shall see in Section 4 the radial solutions of the Dirichlet problem  −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 . (1.2) satisfy an ordinary differential equation like (1.1). ∗ Corresponding author. E-mail addresses: djairo@ime.unicamp.br (D.G. De Figueiredo), jsanchez@userena.cl (J. Sánchez), pedro.ubilla@usach.cl (P. Ubilla). 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.03.061