Strong versus weak local minimizers for the perturbed Dirichlet functional

Abstract.

Let $\Omega \subset{\bf R}^n$ be a bounded domain and $F: \Omega \times {\bf R}^N \to{\bf R}$. In this paper we consider functionals of the form

$I(u):= \int_{\Omega} \left( \frac{1}{2} |Du|^2 + F(x,u) \right) \, dx,$

where the admissible function $u$ belongs to the Sobolev space of vector-valued functions $W^{1,2}(\Omega;{\bf R}^N)$ 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 $1 \le r \le \infty$. 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.