Bound states for a class of quasilinear scalar field equations with potentials vanishing at infinity

We study the existence and non-existence of bound states (i.e., solutions in W1,p(ℝN)W1,p(ℝN)) for a class of quasilinear scalar field equations of the form −Δpu + V(x)|u|p−2 u = a(x)|u|q−2 u, x ∈ ℝN, 1 < p