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 $\mu\in\mathopen{]}0,+\infty\mathclose[$. Specifically, we prove the existence of a critical value $\mu_1> 0$ such that the problem under examination admits: no positive solutions and a continuum of nonnegative solutions compactly supported in $[0,1]$ for $\mu\in\mathopen{]}0,\mu_1\mathclose[$; a unique positive solution of compacton-type for $\mu=\mu_1$; a unique positive solution satisfying Hopf's boundary condition for $\mu\in\mathopen{]}\mu_1,+\infty\mathclose[$.