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

Abstract

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) is asymptotically linear with respect to t at infinity. Ω is a smooth bounded domain in RN and N>4. In this case, 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>0 and M>0,0<F(x,t)=△∫0tf(x,s)ds≤12+θf(x,t)tuniformly a.e. x∈Ω and ∀|t|≥M, which is important in applying the mountain pass theorem. By a variant version of the mountain pass theorem, we obtain the existence of nontrivial solutions to the above problem under suitable assumptions of f(x,t), which generalizes and improves the results in Liu and Wang (2007) [12] and An and Liu (2008) [13].