Superlinear Schrodinger-Kirchhoff type problems involving the fractional p-Laplacian and critical exponent

This paper concerns the existence and multiplicity of solutions for the Schrodinger-Kirchhoff type problems involving the fractional p-Laplacian and critical exponent. As a particular case, we study the following degenerate Kirchhoff-type nonlocal problem: parallel to u parallel to((theta-1)p)(lambda)[lambda(-Delta)(p)(s)u + V(x)vertical bar u vertical bar(p-2)u] = vertical bar u vertical bar(ps)*(-2)u + f(x, u) in RN, parallel to u parallel to(lambda) = (lambda integral(R)integral(2N)vertical bar u(x) - u(y)vertical bar(p)/vertical bar x - y vertical bar(N+ps) dxdy + integral V-RN(x)vertical bar u vertical bar(p)dx)(1/p) where (-Delta)(p)(s) is the fractional p-Laplacian with 0 < s < 1 < p < N/s, p(s)* = Np/(N - ps) is the critical fractional Sobolev exponent, lambda > 0 is a real parameter, 1 < theta <= p(s)*/p, and f : R-N x R -> R is a Caratheodory function satisfying superlinear growth conditions. For theta is an element of (1, p(s)*/p), by using the concentration compactness principle in fractional Sobolev spaces, we show that if f(x, t) is odd with respect to t, for any m is an element of N+ there exists a Lambda(m) > 0 such that the above problem has m pairs of solutions for all lambda is an element of (0, Lambda(m)]. For theta = p(s)*/p, by using Krasnoselskii's genus theory, we get the existence of infinitely many solutions for the above problem for lambda large enough. The main features, as well as the main difficulties, of this paper are the facts that the Kirchhoff function is zero at zero and the potential function satisfies the critical frequency inf(x is an element of R) V(x) = 0. In particular, we also consider that the Kirchhoff term satisfies the critical assumption and the nonlinear term satisfies critical and superlinear growth conditions. To the best of our knowledge, our results are new even in p-Laplacian case.