Quasilinear equations with dependence on the gradient

Author(s)

De Figueiredo, Djairo G. ^[1] ; Sanchez, Justino ^{[2, 3]} ; Ubillac, Pedro ^[2]

Affiliation

^[1] Univ Estadual Campinas, IMECC, BR-13081970 Campinas, SP - Brazil

^[2] Univ Santiago Chile, Dept Matemat & CC, Santiago - Chile

^[3] Univ La Serena, Dept Matemat, La Serena - Chile

• Document type: Journal article
• Source: NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS; v. 71, n. 10, p. 4862-4868, NOV 15 2009.
• Web of Science Citations: 6

Abstract

We discuss the existence of positive solutions of the problem - (q(t)phi(u'(t)))' = f (t, u(t), u'(t)) for t is an element of (0, 1) and u(0) = u(1) = 0. where the nonlinearity f satisfies a superlinearity condition at 0 and a local superlinearity condition at +infinity. This general quasilinear differential operator involves a weight q and a main differentiable part phi which is not necessarily a power. Due to the superlinearity off 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(vertical bar del u vertical bar)del u) = f(vertical bar x vertical bar, u, vertical bar del u vertical bar) in r(1) < vertical bar x vertical bar < r(2), u(x) = 0 on vertical bar x vertical bar = R(1) and vertical bar x vertical bar = R(2). (C) 2009 Elsevier Ltd. All rights reserved. (AU)