Strong versus weak local minimizers for the perturbed Dirichlet functional

Let be a bounded domain and . In this paper we consider functionals of the form where the admissible function belongs to the Sobolev space of vector-valued functions and is such that the integral on the right is well defined. We state and prove a sufficiency theorem for local minimizers of where . The exponent is shown to depend on the dimension and the growth condition of and an exact expression is presented for this dependence. We discuss some examples and applications of this theorem. Received: 20 July 2000 / Accepted: 7 June 2001 / Published online: 19 October 2001