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We study an eigenvalue problem involving a fully anisotropic ellipticdifferential operator in arbitrary Orlicz-Sobolev spaces. The relevantequations are associated with constrained minimization problems for integralfunctionals depending on the gradient of competing functions through generalanisotropic $N$-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 somenew aspects of the theory of anisotropic Orlicz-Sobolev spaces.

An eigenvalue problem for the anisotropic $\Phi$-Laplacian