  The study of fourth order partial differential equations has flourished in
the last years, however, a $p(\cdot)$-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 $f$.
