Existence of positive solution for Kirchhoff type problem with critical discontinuous nonlinearity

In this paper we are concerned with existence of positive solution to the class of nonlinear problems of the Kirchhoff type given by Lǫ(u) = H(u− β)f(u) + u 2 ∗ −1 in R , u ∈ H(R ) ∩W 2, q q−1 (R ), 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 Lǫ(u) := [ M ( 1 ǫN−2 ˆ