Let \(f(\xi)\in C^ 2(R^ n)\) be a real-valued strictly convex function such that \[ (1)\quad m| \xi |^ p\leq f(\xi)\leq M(1+| \xi |^ q)\quad for\quad all\quad \xi \in R^ n. \] Existence and regularity of a minimizer of the functional \[ F(u)=\int_{\Omega}f(Du(x))dx \] have been investigated when \(p=q\) and some suitable condition on the growth of the second derivatives of f are given. Recently the interest on the regularity of minimizers of such functionals where \(p\neq q\) has been pointed out also in connection with some problems in nonlinear elasticity. In this paper the author proposes an approach to the local regularity when the condition (1) is satisfied with \(p\neq q\) and also the second derivatives of f satisfy a non-standard condition.