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We establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic equations involving the anisotropic p ! ðÁÞ-Laplace operator, on a bounded domain with smooth boundary. We work on the anisotropic variable exponent Sobolev spaces and our main tool is the symmetric mountain-pass theorem of Ambrosetti and Rabinowitz.

The study of fourth order partial differential equations has flourished in the last years, however, a p(·)biharmonic problem with no-flux boundary condition has never been considered before, not even for constant p. This is an important step further, since surfaces that are impermeable to some contaminants are appearing quite often in nature, hence the significance of such boundary condition. By relying on several variational arguments, we obtain the existence and the multiplicity of weak solutions to our problem. We point out that, although we use a mountain pass type theorem in order to establish the multiplicity result, we do not impose an Ambrosetti-Rabinowitz type condition, nor a symmetry condition, on our nonlinearity f .

Abstract. We analyze a class of quasilinear elliptic problems involving a p(·)-Laplace-type operator on a bounded domain Ω ⊂ R N , N ≥ 2, and we deal with nonlinear conditions on the boundary. Working on the variable exponent Lebesgue-Sobolev spaces, we follow the steps described by the "fountain theorem" and we establish the existence of a sequence of weak solutions. Mathematics Subject Classification (2010