Multivalued elliptic operators with nonstandard growth

The paper is devoted to the Dirichlet problem for monotone, in general multivalued, elliptic equations with nonstandard growth condition. The growth conditions are more general than the well-known p(x) growth. Moreover, we allow the presence of the so-called Lavrentiev phenomenon. As consequence, at least two types of variational settings of Dirichlet problem are available. We prove results on the existence of solutions in both of these settings. Then we obtain several results on the convergence of certain types of approximate solutions to an exact solution.