Sign-changing saddle point

In this paper, the Saddle-point theorems are generalized to a new version by showing that there exists a “sign-changing” saddle point besides zero. The abstract result is applied to the semilinear elliptic boundary value problem − u = f (x,u) in , u=0 on and the Schrödinger equation   − u + V (x)u = f (x, u), x ∈ RN, u(x) →0 as|x| → ∞, where ⊂ RN is a bounded domain with smooth boundary ; the Schrödinger operator − + V has both eigenvalues and essential spectrum. The asymptotically linear case is considered which permits double resonance to be happened. Some existence results of sign-changing solutions are established.