  In this paper we consider minimizers for nonlocal energy functionals
generalizing elastic energies that are connected with the theory of
peridynamics \cite{Silling2000} or nonlocal diffusion models \cite{Rossi}. We
derive nonlocal versions of the Euler-Lagrange equations under two sets of
growth assumptions for the integrand. Existence of minimizers is shown for
integrands with joint convexity (in the function and nonlocal gradient
components). By using the convolution structure we show regularity of solutions
for certain Euler-Lagrange equations. No growth assumptions are needed for the
existence and regularity of minimizers results, in contrast with the classical
theory.
