Uniqueness of positive and compacton-type solutions for a resonant quasilinear problem

Abstract

We study a one-dimensional p-Laplacian resonant problem with p-sublinear terms and depending on a positive parameter. By using quadrature methods we provide the exact number of positive solutions with respect to µ ∈ ]0, +∞[. Specifically, we prove the existence of a critical value µ 1 > 0 such that the problem under examination admits: no positive solutions and a continuum of nonnegative solutions compactly supported in [0, 1] for µ ∈ ]0, µ 1 [; a unique positive solution of compacton-type for µ = µ 1 ; a unique positive solution satisfying Hopf's boundary condition for µ ∈ ]µ 1 , +∞[.