  We determine the shape which minimizes, among domains with given measure, the
first eigenvalue of the anisotropic laplacian perturbed by an integral of the
unknown function. Using also some properties related to the associated \lq\lq
twisted\rq\rq problem, we show that, this problem displays a \emph{saturation}
phenomenon: the first eigenvalue increases with the weight up to a critical
value and then remains constant.
