Regularity results for elliptic equations and systems with variable growth exponent

We present a review on some regularity results I obtained in the last 10 years for elliptic equations whose prototype is the p(x)-Laplacian; they can be interpreted as the Euler-Lagrange equations of integral functionals appearing in the mathematical modelling of strongly anisotropic materials. Under suitable continuity assumptions on the function p, the results I'm going to present include: - Hoelder continuity results in the scalar case (also for the obstacle problem) - Calderon-Zygmund estimates for a class of obstacle problem with variable growth exponent - global regularity and stability of solutions to elliptic equations with non-standard growth - Lipschitz estimates for systems (thus in the vectorial setting) with ellipticity conditions at infinity