A Class of Kirchhoff-Type Problems Involving the Concave–Convex Nonlinearities and Steep Potential Well.

This paper is concerned with the following Kirchhoff-type problem: - a + b ∫ R 3 | ∇ u | 2 d x ▵ u + λ V (x) u = g (x , u) + f (x , u) in R 3 , u ∈ H 1 (R 3) , where a, b and λ are real positive parameters. The nonlinearity g (x , u) + f (x , u) may involve a combination of concave and convex terms. By assuming that V represents a potential well with the bottom V - 1 (0) , under some suitable assumptions on f , g ∈ C (R 3 × R , R) , we obtain a positive energy solution u b , λ + via combining the truncation technique and get the asymptotic behavior of u b , λ + as b → 0 and λ → + ∞ . Moreover, we also give the existence of a negative energy solution u b , λ - via Ekeland variational principle. [ABSTRACT FROM AUTHOR]