COMBINED EFFECTS FOR FRACTIONAL SCHRODINGER-KIRCHHOFF SYSTEMS WITH CRITICAL NONLINEARITIES

In this paper, we investigate the existence of solutions for critical Schrodinger-Kirchhoff type systems driven by nonlocal integro-differential operators. As a particular case, we consider the following system: {M ([u,v)](s,p)(p) + parallel to(u,v)parallel to(p)(p,V)) ((-Delta)(p)(s)u + V(x) vertical bar u vertical bar(p-2)u) = lambda H-u(x,u,v) + alpha/p*s vertical bar v vertical bar(beta) vertical bar u vertical bar(alpha-2)u in R-N {M ([u,v)](s,p)(p) + parallel to(u,v)parallel to(p)(p,V)) ((-Delta)(p)(s)u + V(x) vertical bar u vertical bar(p-2)u) = lambda H-v(x,u,v) + beta/p*(s)vertical bar u vertical bar(alpha) vertical bar v vertical bar(beta-2)v in R-N, where (-Delta)(p)(s) is the fractional p-Laplace operator with 0 < s <1 < p < N/s, alpha,beta > 1 with alpha+ beta = p*(s), M : R-0(+) -> R-0(+)) is a continuous function, V : R-N -> R+ is a continuous function, lambda > 0 is a real parameter. By applying the mountain pass theorem and Ekeland's variational principle, we obtain the existence and asymptotic behaviour of solutions for the above systems under some suitable assumptions. A distinguished feature of this paper is that the above systems are degenerate, that is, the Kirchhoff function could vanish at zero. To the best of our knowledge, this is the first time to exploit the existence of solutions for fractional Schrodinger-Kirchhoff systems involving critical nonlinearities in R-N.