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We study energy functionals obtained by adding a possibly discontinuouspotential to an interaction term modeled upon a Gagliardo-type fractionalseminorm. We prove that minimizers of such non-differentiable functionals arelocally bounded, H\"older continuous, and that they satisfy a suitable Harnackinequality. Hence, we provide an extension of celebrated results of M.Giaquinta and E. Giusti to the nonlocal setting. To do this, we introduce aparticular class of fractional Sobolev functions, reminiscent of thatconsidered by E. De Giorgi in his seminal paper of 1957. The flexibility ofthese classes allows us to also establish regularity of solutions to rathergeneral nonlinear integral equations.

Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes