Existence of nontrivial solutions for fourth-order asymptotically linear elliptic equations

In this paper, we discuss the following fourth-order semilinear elliptic problem {Δ2u+cΔu=f(x,u),x∈Ω,u=Δu=0,x∈∂Ω, where f(x,t)f(x,t) is asymptotically linear with respect to tt at infinity. ΩΩ is a smooth bounded domain in RNRN and N>4N>4. In this case, f(x,t)f(x,t) does not satisfy the following Ambrosetti–Rabinowitz type condition (see Ambrosetti and Rabinowitz (1973)  [9]), for short, which is called the (AR) condition, that is, for some θ>0θ>0 and M>0M>0,0