Positive solutions of a focal problem for one-dimensional p-Laplacian equations

This paper mainly deals with the existence and multiplicity of positive solutions for the focal problem involving both the p-Laplacian and the first order derivative: {((u')(p-1))' + f (t, u, u') = 0, t is an element of(0, 1), u(0) = u'(1) = 0. The main tool in the proofs is the fixed point index theory, based on a priori estimates achieved by using Jensen's inequality and a new inequality. Finally the main results are applied to establish the existence of positive symmetric solutions to the Dirichlet problem: {(|u'|(p-2)u')' + f (u, u') = 0, t is an element of (-1, 0) boolean OR (0, 1), u(-1) = u(1) = 0. (C) 2011 Elsevier Ltd. All rights reserved.