On the Unboundedness of the Number of Solutions of a Dirichlet Problem

We confirm a conjecture raised by Lazer and McKenna on the number of Dirichlet solutions of the equation u 00 + f(u) = s sin(t) + h(t) in [0; ß], where the nonlinear function f(u) satisfies \Gamma1 ! f 0 (\Gamma1) ! f 0 (1) = 1. Our result asserts that given any positive integer N , there exists a real number s N such that for all s ? s N there are at least N Dirichlet solutions. AMS(MOS) Subject Classification. Primary 34B15. Secondary 35J25. Key Words and Phrases. Nonlinear boundary value problem, variation index, shooting method, multiplicity of solutions. Proposed Running Head. Unbounded Number of Solutions This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U. S. Department of Energy, under Contract W-31-109-Eng-38. 1 Introduction In an informative article [1], Lazer and McKenna proposed a modified mathematical model for the onset of large-amplitude oscillations in suspension bridges by wind with specific velocitie..