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-Δu=f(x,u)inΩ,u=0on∂Ωand the Schrödinger equation -Δu+Vλ(x)u=f(x,u),x∈RN,u(x)→0as|x|→∞,where Ω⊂RN is a bounded domain with smooth boundary ∂Ω; the Schrödinger operator -Δ+Vλ 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.