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In this paper, we study the following fractional Schrödinger Kirchhoff type problem (Qǫ) { Lǫu = K(x)f(u) in R , u ∈ H(R), where Lǫ is a nonlocal operator defined by L s ǫu = M ( 1 ǫ3−2s ̈ R×R |u(x)− u(y)| |x− y|3+2s dx dy + 1 ǫ ˆ R V (x)u dx ) [ǫ(−∆)u+ V (x)u], ǫ is a small positive parameter, 3 4 < s < 1 is a fixed constant, the operator (−∆) is the fractional Laplacian of order s, M , V , K and f are continuous functions. Under proper assumptions on M , V , K and f , we prove the existence and concentration phenomena of solutions of the problem (Qǫ). With minimax theorems and the Ljusternik–Schnirelmann theory, we also obtain multiple solutions of problem (Qǫ) by employing the topology of the set where the potentials V (x) attains its minimum and K(x) attains its maximum.

The existence and concentration behavior of ground state solutions for fractional Schrödinger–Kirchhoff type equations