In this paper we are concerned with existence of positive solution to the class of nonlinear problems of the Kirchhoff type given by L ǫ (u) = H(u − β)f (u) + u 2 * −1 in R N , u ∈ H 1 (R N) ∩ W 2, q q−1 (R N), where N ≥ 3, q ∈ (2, 2 *), ǫ, β > 0 are positive parameters, f : R → R is a continuous function, H is the Heaviside function, i.e., H(t) = 0 if t ≤ 0, H(t) = 1 if t > 0 and L ǫ (u) := M 1 ǫ N −2ˆR N |∇u| 2 dx + 1 ǫ NˆR N V (x)|u| 2 dx [−ǫ 2 ∆u + V (x)u]. The function M is a general continuous function. The function V is a positive potential that satisfies following hypothesis: or V satisfies the Palais-Smale condition or there is a bounded domain Ω in R N such that V has no critical point in ∂Ω. Here we use a suitable truncation to apply a version of the penalization method of Del Pino and Felmer [16] combined with the Mountain Pass Theorem for locally Lipschitz functional.

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