A concave-convex Kirchhoff type elliptic equation involving the fractional p-Laplacian and steep well potential

In this paper, we study the following Schrodinger-Kirchhoff type equation: (a + b integral integral(R2N) vertical bar u(x) - u(y)vertical bar(p)/vertical bar x - y vertical bar(N+ps) dx dy) (-Delta)(p)(s)u + lambda V(x) vertical bar u vertical bar(p-2)u = h(x)vertical bar u vertical bar(r-2) u + g(x)vertical bar u vertical bar(q-2)u, x is an element of R-N, where s is an element of (0, 1 ),2 <= p < infinity, N > ps, 1 < q < p < r < p(s)* and alpha,lambda > 0, b >= 0 are three real parameters. By using the mountain pass theorem and a variant version of Ekeland's variational principle in Zhong [A generalization of Ekeland's variational principle and application to the study of relation between the weak P.S. condition and coercivity. Nonlinear Anal. 1997;29:1421-1431], we get some results. Firstly, we obtain a positive energy solution u(b,lambda)(+) by a truncated functional and discuss their asymptotical behavior for lambda -> + infinity when p < r < 2p and b lies in suitable range. Secondly, for all b > 0, we obtain a positive energy solution u(lambda)(+) and discuss their asymptotical behavior for lambda -> + infinity when 2p <= r < P-s*, where P-s* = N-p/N-sp. Moreover, we obtain other asymptotical behavior of positive energy solution when p < r < p(s)*. Finally, we get a negative energy solution and the non-existence of the non-trivial solutions.