We investigate the existence and the multiplicity of weak solutions for fourth-order problems involving the Leray-Lions type operators in variable exponent spaces. The proofs of our main results use variational methods and combine three critical theorems of Bonanno and Marano (Appl. Anal. 89 (2010), 1-10).

In this paper, we deal with the fractional p(.,.)-Laplacian problem with nonlocal Neumann boundary conditions. Applying the Ekeland principle and the variational method, under appropriate assumptions, we show that the above problem has nontrivial weak solutions.