ANISOTROPIC QUASILINEAR ELLIPTIC EQUATIONS WITH VARIABLE EXPONENT

We study some anisotropic boundary value problems involv- ing variable exponent growth conditions and we establish the existence and multiplicity of weak solutions by using as main argument critical point theory. where p(x) > 1 is a continuous function. Such kind of functionals are men- tioned, for instance, in the work of Ruzicka (18) where they are used to model an electrorheological fluid. They correspond to the so called a p(x)-Laplace operator which is described by the formula �p(x)u = div(橲 uj p(x)−2 r u): However, if we seek for the model of an inhomogeneous material which has a different behavior on each direction we note that the above energy is not adequate. In this new case an appropriate form for energetic functionals can be described by the formula (2)