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

In this paper, we discuss the following fourth-order semilinear elliptic problem {Delta(2)u + c Delta u = f(x, u), x is an element of Omega, u = Delta u = 0, x is an element of partial derivative Omega, where f(x, t) is asymptotically linear with respect to t at infinity. Omega is a smooth bounded domain in R-N 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 theta > 0 and M > 0, 0 < F(x, t) (sic) integral(t)(0)f(x, s)ds <= 1/2 + theta f(x, t)t uniformly a.e. x is an element of Omega and for all vertical bar t vertical bar >= 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]. (C) 2013 Elsevier Ltd. All rights reserved.