multiplicity result for a two point boundary value problem involving a fourth order equation

The existence of solutions for fourth-order elastic beam equation is discussed. in fact, employing a consequence of the local minimum theorem due to bonanno and mountain pass theorem the existence results for the problem under algebraic conditions with the classical ambrosetti-rabinowitz (ar) condition on the nonlinear term are established. furthermore,by combining two algebraic conditions on the nonlinear term using two consequences of the local minimum theorem due to bonanno the existence of two solutions is ensured, applying the mountain pass theorem given by pucci and serrin the existence of third solution for our problem is established.