An eigenvalue problem for quasilinear systems

摘要

The paper deals with the existence of positive solutions for the n-dimensional 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) = (phi(u(1)), . . . , phi(u(n))), where u = (u(1), . . . , u(n)), and phi covers the two important cases phi(u) = u and phi(u) = vertical bar u vertical bar(p - 2)u, 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(vertical bar vertical bar u vertical bar vertical bar) (->) (0) f(i) (u)/phi(vertical bar vertical bar u vertical bar vertical bar), f(infinity)(i) = lim(vertical bar vertical bar u vertical bar vertical bar -> infinity) f(i) (u)/phi(vertical bar vertical bar u vertical bar vertical bar), 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 theorems in a cone.