Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems

In this paper, we prove a mountain pass theorem in order intervals in which the position of the mountain pass point is given precisely in terms of the order structure. By using this result and constructing special flows, we deal with the existence of multiple solutions and sign-changing solutions for the following classes of elliptic Dirichlet boundary value problems: (1) nonlinear terms have concave property near zero and have superlinear but subcritical growth at infinity; (2) nonlinear terms are of the form h(x)f(u), with h(x) changing sign; (3) the asymptotically linear case. We obtain several new existence results of nodal solutions and give more comparable relations among the positive, negative and sign-changing solutions obtained. Our method is set up in an abstract setting and should be useful in other problems.