Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type

In this paper by using the minimal principle and Morse theory, weprove the existence of solutions to the following Kirchhoffnonlocal fractional equation:{eqnarray}& M (∫R^n× R^n |u (x) - u (y)|^2 K (x - y) d x d y ) (- Δ)^s u = f (x, u (x)), in\;\; Ω,\& u = 0, in\;\; R^n Ω,{eqnarray}where (- Δ)ˢ is the fractional Laplace operator, s ∈(0, 1) is a fix, Ω an open bounded subset ofRⁿ, $n > 2 s$, with Lipschitz boundary, f: Ω× R → R Carathéodory function and M :R⁺ → R⁺ is a function that satisfysome suitable conditions.