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 integral(3)(R) vertical bar del u(vertical bar 2) dx) Delta u +lambda V(x)u = g(x, u) + f (x, u) in R-3, u is an element of H-1(R-3), where a, b and lambda 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 is an element of C(R-3 x R, R), we obtain a positive energy solution u(b,lambda)(+) via combining the truncation technique and get the asymptotic behavior of u(b,lambda)(+) as b -> 0 and lambda -> + infinity. Moreover, we also give the existence of a negative energy solution u(b,lambda)(-) via Ekeland variational principle.