Semilinear equations in analysis and mechanics: some history and recent progress

Starting with the classical result of Ambrosetti and Prodi, I will review the history of an area that can be summarized as: asymmetric nonlinearities with positive loads have many solutions... the more asymmetric the nonlinearity, the more solutions. We shall look at the history for semilinear elliptic equations, and also some applications to simple mechanical systems such as cable-suspended masses.