  We study an eigenvalue problem involving a fully anisotropic elliptic
differential operator in arbitrary Orlicz-Sobolev spaces. The relevant
equations are associated with constrained minimization problems for integral
functionals depending on the gradient of competing functions through general
anisotropic $N$-functions. In particular, the latter need neither be radial,
nor have a polynomial growth, and are not even assumed to satisfy the so called
$\Delta_2$-condition. The resulting analysis requires the development of some
new aspects of the theory of anisotropic Orlicz-Sobolev spaces.
