Exact multiplicity results for quasilinear boundary-value problems with cubic-like nonlinearities

We consider the boundary-value problem -(phi(p)(u'))' = lambda f(u) in (0, 1) u(0) = u(1) = 0, where p > 1, lambda > 0 and phi(p)(x) = vertical bar x vertical bar(p -2)x. The nonlinearity f is cubiclike with three distinct roots 0 = a < b < c. By means of a quadrature method, we provide the exact number of solutions for all lambda > 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 di ff erent in the case p > 2.