Nonlinear eigenvalue problems for quasilinear systems

The paper deals with the existence of positive solutions for the quasilinear system (Phi(u'))'+lambda h(t)f(u) = 0, 0 < t < 1 with the boundary condition u(0) = u(1) = 0. The vector-valued function Phi is defined by Phi(u) = (q(t)psi(p(t)u(1)),..., q(t)psi(p(t)u(n))), where u = (u(1),...,u(n)), and psi covers the two important cases psi(u) = u and psi(u) = vertical bar u vertical bar(p-2)u(1) p > 1, h(t) = diag[h(1) (t),..., h(n)(t)] and f(u) = (f(1)(u),..., f(n)(u)). Assume that f(i) and h(i) are nonnegative continuous. For u = (u(1),...,u(n)), let f(0)(i) = lim(parallel to u parallel to -> 0) f(i)(u)/psi(parallel to u parallel to), f(infinity)(i) = lim(parallel to u parallel to -> infinity) f(i)(u)/psi(parallel to u parallel to), (i=1,...,n), f(0) = max{f(0)(1),...,f(0)(n)} and f(infinity) = max{f(infinity)(1),...,f(infinity)(n)}. We prove that the boundary value problem has a positive solution, for certain finite intervals of lambda, if one of f(0) and f(infinity) is large enough and the other one is small enough. Our methods employ fixed-point theorem in a cone. (c) 2005 Elsevier Ltd. All rights reserved.