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A perturbed eigenvalue problem on general domains

The perturbed eigenvalue problem −∆u − ∆pu = λV (x)u, with p ∈ (1, N) \ {2} and V a weight function which takes nonnegative values and may have singular points, is studied in an Orlicz–Sobolev setting on general open sets from R with N ≥ 3. The analysis of these problems leads to a full characterization of the set of parameters λ for which the problem possesses nontrivial solutions as being an unbounded open interval.