Exactness of the number of positive solutions to a singular quasilinear problem

Abstract

<p>We study the exact multiplicity of positive solutions to the one-dimensional Dirichlet problem</p> <pre>-(|u′|^p-2 u′)′ = λu^s-1 - μu^r-1 in ]0, 1[<br> u(0) = u(1) = 0,</pre> <p>where r ∈ ]0, 1[, p ∈ ]1, +∞[, r < s < p and λ, μ ∈ ]0, +∞[. We shed light, in particular, on the case r ∈ ]0, min{s, p/(p + 1)}[, completely determining the bifurcation diagram and solving some related open problems. Our approach relies upon quadrature methods.</p>