Positive solutions for the p-Laplacian in annuli

Abstract

We consider the Dirichlet problem for the equation -Δ_pu = q(|x|)f(u) in an annulus Ω ⊂ Rⁿ, n ≥ 1, where Δ_pu = div(|∇u|^p−2∇u) is the p-Laplacian operator, p > 1. With no assumption on the behaviour of the nonlinearity f either at zero or at infinity, we prove existence and localization of positive radial solutions for this problem by applying Schauder's fixed-point theorem. Precisely, we show the existence of at least one such solution each time the graph of f passes through an appropriate tunnel. So, it is easy to exhibit multiple, or even infinitely many, positive solutions. Moreover, upper and lower bounds for the maximum value of the solution are obtained. Our results are easily extended to the exterior of a ball, when n > p.