Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities

In this paper, we investigate the existence of solutions for critical Schrödinger–Kirchhoff type systems driven by nonlocal integro–differential operators. As a particular case, we consider the following system: [see formula in PDF] where (– Δ ) s p is the fractional p –Laplace operator with 0 < s < 1 < p < N / s , α , β > 1 with α + β = p * s , M : ℝ + 0 → ℝ + 0 is a continuous function, V : ℝ N → ℝ + is a continuous function, λ > 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 Schrödinger–Kirchhoff systems involving critical nonlinearities in ℝ N .