Z. Angew. Math. Phys. (2016) 67:73 c  2016 Springer International Publishing DOI 10.1007/s00033-016-0668-5 Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP Existence of one weak solution for p(x)-biharmonic equations with Navier boundary conditions S. Heidarkhani, G. A. Afrouzi, S. Moradi, G. Caristi and Bin Ge Abstract. We study the existence of at least one weak solution for a class of elliptic Navier boundary value problems involving the p(x)-biharmonic operator. Our technical approach is based on variational methods. In addition, an example to illustrate our results is given. Mathematics Subject Classification. 35J20, 35J60. Keywords. Navier condition, p(x)-biharmonic operator, Existence result, Variational methods. 1. Introduction The aim of this paper is to establish the existence of at least one weak solution for the following Navier boundary value problem  Δ 2 p(x) u = f (x, u(x)), x ∈ Ω, u =Δu =0, x ∈ ∂ Ω (P f ) where Ω ⊂ R N (N ≥ 2) is a bounded domain with boundary of class C 1 ,f ∈ C 0 (Ω × R), p(·) ∈ C 0 (Ω) with max  2, N 2  <p − := inf x∈ Ω p(x) ≤ p + := sup x∈ Ω p(x) and Δ 2 p(x) u := Δ(|Δ| p(x)−2 Δu) is the operator of fourth order called the p(x)-biharmonic operator, which is a natural generalization of the p-biharmonic operator (where p> 1 is a constant). The topic of function spaces with variable exponents has undergone an impressive development over the last decades. Such problems arise from the study of electrorheological fluids, image processing and the theory of nonlinear elasticity. The operator Δ p(x) u := div(|∇u| p(x)−2 ∇u) is called p(x)-Laplacian, which becomes p-Laplacian when p(x) ≡ p (a constant). The study of various mathematical problems with variable exponent has received considerable attention in recent years. These problems are interesting in applications (see, for example, [28]) and raise many difficult mathematical problems. In recent years, the study of problems involving biharmonic, p-biharmonic and p(x)-biharmonic op- erators has been widely approached. For background and recent results, we refer the reader to [1, 3–6, 9– 16, 20, 23–26] and the references therein for details. For example, in [9, 13] using variational methods and Supported by the NNSF of China (No. 11201095), the Youth Scholar Backbone Supporting Plan Project of Harbin Engineering University, the Fundamental Research Funds for the Central Universities (No. 2016), Postdoctoral research startup foundation of Heilongjiang (No. LBH-Q14044), the Science Research Funds for Overseas Returned Chinese Scholars of Heilongjiang Province (No. LC201502). 0123456789().: V,-vol