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We prove that the quasilinear equation − p u = λV |u| p−2 u + g(x, u), with g subcritical and p-superlinear at 0 and at infinity, admits a nontrivial weak solution u ∈ W 1,p 0 (Ω) for any λ ∈ R. A minimax approach, allowing also an estimate of the corresponding critical level, is used. New linking structures, associated to certain variational eigenvalues of − p u = λV |u| p−2 u, are recognized, even in absence of any direct sum decomposition of W 1,p 0 (Ω) related to the eigenvalue itself.