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An eigenvalue problem for the anisotropic $Φ$-Laplacian

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 $Δ_2$-condition. The resulting analysis requires the development of some new aspects of the theory of anisotropic Orlicz-Sobolev spaces.