Integral representation and Γ-convergence of variational integrals with

We study the integral representation properties of limits of sequences of integral functionals like $\int f(x,Du)\,{\rm d}x$ under nonstandard growth conditions of (p,q)-type: namely, we assume that $$ \vert z\vert^{p(x)}\leq f(x,z)\leq L(1+\vert z\vert^{p(x)})\,. $$ Under weak assumptions on the continuous function p(x), we prove Γ-convergence to integral functionals of the same type. We also analyse the case of integrands f(x,u,Du) depending explicitly on u; finally we weaken the assumption allowing p(x) to be discontinuous on nice sets