Multiplicity Results for Classes of Infinite Positone Problems

We study positive solutions to the singular boundary value problem { -Delta(p)u = lambda(f(u)/u beta) in Omega u = 0 on partial derivative Omega, where Delta(p)u = div (vertical bar del u vertical bar(p-2)del u), p > 1, lambda > 0, beta is an element of (0, 1) and Omega is a bounded domain in R-N, N >= 1. Here f : [0, infinity) -> (0, infinity) is a continuous nondecreasing function such that lim(u ->infinity) f(u)/u(beta+p-1) = 0. We establish the existence of multiple positive solutions for certain range of lambda when f satisfies certain additional assumptions. A simple model that will satisfy our hypotheses is f(u) = e(alpha u/alpha+u) for alpha >> 1. We also extend our results to classes of systems when the nonlinearities satisfy a combined sublinear condition at infinity. We prove our results by the method of sub-supersolutions.