The Sobolev class where a weak solution is a local minimizer

The aim of this paper is to propose some results which we hope could contribute to understand better Lavrentiev’s phenomenon for energy integrals as in (1.1) under some p,q-growth conditions as in (1.2); in fact, we expect that Lavrentiev’s phenomenon does not occur if the quotient q=p is not too large in dependence of n, for instance, as in the cases – either scalar or vectorial ones – that we consider in this manuscript.