Exact Multiplicity Results for Quasilinear Boundary-value Problems with Cubic-like Nonlinearities

Abstract

We consider the boundary-value problem -(φp(u'))' = λf(u) in (0,1) u(0) = u(1) = 0, where p > 1, λ > 0 and φ_p(x) = |x|p-2x. The nonlinearity ƒ is cubic-like with three distinct roots 0 = α < b < c. By means of a quadrature method, we provide the exact number of solutions for all λ > 0. This way we extend a recent result, for p = 2, by Korman et al. [17] to the general case p > 1. We shall prove that when 1 < p ≤ 2 the structure of the solution set is exactly the same as that studied in the case p = 2 by Korman et al. [17], and strictly different in the case p > 2.