In this paper, we study the existence, nonexistence and L∞−solutions to the following p−Kirchhoff equation −M(||∇u||p)∆pu + V(x)|u|p−2u = f(x, u), x ∈ R^N ,where M(t) = a + btpτ , t ≥ 0, a > 0, b, τ ≥ 0, 1 < p < N, and V (x) is a continuous and nonnegative function in RN . Under suitable conditions on V (x) and f(x, u), the existence of infinitely many bounded solutions and the nonexistence of solutions to Eq.(0.1) are investigated.