Existence of n solutions and/or positive solutions to a semipositone elastic beam equation

The existence of n solutions and/or positive solutions is established for a nonlinear fourth-order two-point boundary value problem where n is an arbitrary natural number and the nonlinear term has a nonpositive lower bound. In mechanics, the problem describes deformation of the elastic beam whose two ends are simply supported. The main results show that the problem has at least n solutions and/or positive solutions provided that the “heights” of the nonlinear term are appropriate on some bounded sets.