Nonlinear Analysis 59 (2004) 319 – 333 www.elsevier.com/locate/na A p-Laplacian problem with a multi-point boundary condition Marta García-Huidobro a , Rául Manásevich b, 1 , Ping Yan c , Meirong Zhang c , ∗, 2 a Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile b Departamento de Ingeniería Matemática, Universidad de Chile and Centro de Modelamiento Matemático, UMR2071 CNRS-UChile, Casilla 170, Correo 3, Santiago, Chile c Department of Mathematical Sciences, Tsinghua University, Haidian District, Beijing 100084, China Received 7 May 2004; accepted 19 July 2004 Abstract In this paper we will use the degree theory to give an existence result to the boundary value problem of nonlinear perturbations of the scalar p-Laplacian with the multi-point boundary condition involving the Lebesgue-Stieltjes integral. Among the boundary conditions considered in this paper, we will find also the optimal bounds on the growth of the nonlinear perturbations in our existence result.  2004 Elsevier Ltd. All rights reserved. Keywords: p-Laplacian; Multi-point boundary condition; Lebesgue-Stieltjes integral; Degree theory; Eigenvalue; L p norms 1. Introduction In recent years, the boundary value problems (BVPs) of nonlinear differential equations with the following multi-point boundary condition (over the interval ∗ Corresponding author. Tel.: +86-10-6277-2863; fax: +86-10-6278-1785. E-mail addresses: mgarcia@mat.puc.cl (M. García-Huidobro), manasevi@dim.uchile.cl (R. Manásevich), pyan@math.tsinghua.edu.cn (P. Yan), mzhang@math.tsinghua.edu.cn (M. Zhang). 1 Supported by Fondap-Conicyt of Chile. 2 Supported by the NNSFC (no. 10325102), TRAPOYT-M.O.E. (2001) and the National 973 Project (no. G1999075108) of China. 0362-546X/$ - see front matter  2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.07.015