We are concerned with a nonlinear elliptic equation, involving a Kirchhoff type nonlocal term and a potential V(x), on ℝ3. As is well known that, even in \(H^1_r(\mathbb{R}^3)\) , the nonlinear term is a pure power form of ∣u∣p−1u and V(x) ≢ 1, it seems very difficult to apply the mountain-pass theorem to get a solution (i.e., mountain-pass solution) to this kind of equation for all p ∈ (1, 5), due to the difficulty of verifying the boundedness of the PalaisSmale sequence obtained by the mountain-pass theorem when p ∈ (1, 3). In this paper, we find a new strategy to overcome this difficulty, and then get a mountain-pass solution to the equation for all p ∈ (1, 5) and for both V(x) being constant and nonconstant. Also, we find a possibly optimal condition on V(x).
