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Existence of solutions for a p(x)-biharmonic problem under Neumann boundary conditions

In this paper, we consider for a given smooth bounded domain Ω of , the following -biharmonic type problem where λ is a positive parameter, with ξ is a function which satisfies the condition with and is a function. We prove the existence of a continuous family of eigenvalues in a neighbourhood of the origin, under some suitable conditions by using Ekeland's principle and variational method.