부산대학교 도서관

We prove a result of existence and localization of positive solutions of the Dirich- let problem for -Δpu = w(x)f(u) in a bounded domain Ω, where Δp is the p-Laplacian, w is a weight function and the nonlinearity f(u) 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 on 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 on the nonlinearity which provides the existence of a sequence of positive solutions for the problem with arbitrarily large sup norm, but also to an estimate of the first eigenvalue λp(Ω, w) of the p-Laplacian operator with weight w. For w ≡ 1, we compare our lower bound for λp(Ω, 1) with that obtained by means of the Cheeger constant h(Ω). We give a characterization of this constant in terms of the solution of the torsional creep problem -Δpɸp = 1 in Ω with Dirichlet boundary data, which offers a good approximation of the first eigenvalue of the p-Laplacian for p near 1.