On the unboundedness of the number of solutions of a Dirichlet problem

Abstract

IN AN informative article [ 11, Lazer and McKenna proposed a modified mathematical model for the onset of large-amplitude oscillations in suspension bridges by wind with specific velocities. The study was motivated by the inadequacy of older theories to explain the collapse of the Tacoma Narrows Bridge of Seattle in 1941. In the Lazer-McKenna model, the motion of the bridge is, as usual, governed by a system of differential equations, more specifically, semilinear elliptic differential equations, the complexity of which depends on the degree of approximation and simplifications one is willing to accept. One of the new ideas introduced is the asymmetry of the restoring force from a cable, with respect to expansion and compression. The author’s basic assumption is that the cable “strongly resists expansion, but does not resist compression”. The study of elliptic equations involving a nonlinear restoring-force term of this type is still largely unexplored. In the same article, Lazer and McKenna posed many interesting open questions. Some of these have not been answered even in the one-dimensional case, when the elliptic equation becomes a second-order nonlinear ordinary differential equation. In an earlier article [5], we gave a counterexample to their problem 4. In this article, we take up another one of these questions (problem 2) concerning a nonlinear function that grows very fast at infinity. The boundary value problem we are interested in is