Existence of positive solution for the nonlinear Kirchhoff type equations in the half space with a hole

(1.2) −∆u + u = |u|u, x ∈ Ω, u ∈ H 0 (Ω), where 1 < p < 5. When Ω is a bounded domain, by applying the compactness of the embedding H 0 (Ω) →֒ L(Ω), 1 < p < 6, there is a positive solution of (1.2). If Ω is an unbounded domain, we can not obtain a solution for problem (1.2) by using Mountain Pass Theorem directly because the embedding H 0 (Ω) →֒ L(Ω), 1 < p < 6 is not compact. However, if Ω = R, Berestycki–Lions [3] proved that there is a radial positive solution of equation (1.2) by applying the compactness of the embedding H r (R ) →֒ L(R), 2 < p < 6, where H r (R) consists of the radially symmetric functions in H(R). By the Lions’s Concentration-Compactness Principle [13], there