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We study the existence and non-existence of bound states (i.e.,solutions in W1,p(RN)) for a class of quasilinear scalar field equations of the form -Apμ + v(x)|u|p-2u =a(x)|u|q-2u, x ∈ RN, 1 ＜ p ＜ N,when the potentials V(.) ≥ 0 and a(.) decay to zero at infinity.