Positive Solutions for the p-Laplacian and Bounds for its First Eigenvalue

We prove a result of existence and localization Of Positive Solutions of the Dirichlet, problem for -Delta(p)u = w(x)f(u) in a bounded domain Omega, where Delta(p) is the p-Laplacian, w is a weight, function and the nonlinearity f(a) satisfies certain local bounds. As in Previous results in radially symmetric domains by two of the authors, and in contrast with the hypotheses usually made, no asymptotic behavior is assumed oil f. A positive solution is obtained by applying the Schauder Fixed Point Theorem and such approach allows us to construct ordered sub- and super-solutions for the problem, thus producing an iterative method to obtain a positive solution. Our result leads not only to asymptotic conditions oil the nonlinearity which provides the existence of a sequence of positive solutions for the problem with arbitrarily large sup norm, but also to all estimate of the first eigenvalue lambda(p)(Omega, w) of the p-Laplaciau operator with weight w. For w equivalent to 1, we compare our lower bound for lambda(p)(Omega, 1) with that obtained by means of the Cheeger constant h(Omega). We give a characterization of this constant in terms of the solution of the torsional creep problem -Delta(p)phi(p) = 1 in Omega with Dirichlet boundary data, which offers a good approximation of the first eigenvalue of the p-Laplacian for p near 1.