Existence results for Kirchhoff-type superlinear problems involving the fractional Laplacian

In this paper, we study the existence and multiplicity of solutions for Kirchhoff-type superlinear problems involving non-local integro-differential operators. As a particular case, we consider the following Kirchhoff-type fractional Laplace equation:.{M(integral integral(R2N) vertical bar u(x) -u(y)vertical bar(2)/vertical bar x-y vertical bar(N vertical bar 2s) dxdy) (-Delta)(s)u = f(x,u) in Omega, , where (-.)s is the fractional Laplace operator, s. (0, 1), N > 2s, O is an open bounded subset of RN with smooth boundary.O, M : R+ 0. R+ is a continuous function satisfying certain assumptions, and f(x, u) is superlinear at infinity. By computing the critical groups at zero and at infinity, we obtain the existence of non-trivial solutions for the above problem via Morse theory. To the best of our knowledge, our results are new in the study of Kirchhoff-type Laplacian problems.