Strong versus weak local minimizers for the perturbed Dirichlet functional

Let Omega subset of R-n be a bounded domain and F : Omega x R-N --> R. In this paper we consider functionals of the form I(u) := fOmega(1/2\Du\(2) + F(x, u)) dx, where the admissible function u belongs to the Sobolev space of vector-valued functions W-1,W-2 (Omega; R-N) and is such that the integral on the right is well defined. We state and prove a sufficiency theorem for L-r local minimizers of I where 1 less than or equal to r less than or equal to infinity. The exponent r is shown to depend on the dimension n and the growth condition of F and an exact expression is presented for this dependence. We discuss some examples and applications of this theorem.