Existence of Positive Solutions for the Kirchhoff Type Equations Involving General Critical Growth in \(\mathbb{R}\)\(^{N}\)

In this paper, we consider the following nonlinear Kirchhoff type problem − ( a + \(\lambda\) \(\int\) \(\mathbb{R}\)\(^{N}\) |∇u|2dx ) Δu + V (x)u = f(u), x ∈ \(\mathbb{R}\)\(^{N}\), where N ≥ 3, a is a positive constant, \(\lambda\) \(\ge\) 0 is a parameter. Under some sufficient assumptions on V (x) and f(u), the existence of positive solution to the above problem is proved by variational methods and Mountain Pass Theorem. Specially, with the aid of a cut-off function and a monotonic trick, we obtain the boundedness of Palais-smale sequences. In this paper, we consider variable potential V (x) and more general f without Ambrosetti-Rabinowitz condition and the monotonicity of f(u)/u. Thus, our results improve the previous results in the literature.