Multiple solutions of some boundary value problems with parameters

In this paper, we study the existence and multiplicity of nontrivial solutions for the following second-order Dirichlet nonlinear boundary value problem with odd order derivative: -u''(t) + au'( t) + bu(t) = f(t,u( t)) for all t epsilon [0, 1] with u( 0) = u( 1) = 0, where a, b epsilon R-1, f epsilon C-1([0, 1] x R-1, R-1). By using the Morse theory, we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution, two nontrivial solutions and infinitely many solutions, separately. (C) 2010 Elsevier Ltd. All rights reserved.