Abstract. By means of variational methods we investigate existence, non-existence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.

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In this work we study the existence of multiple solutions for the non-homogeneous systemwherefor some s > N . Using variational methods, we prove the existence of at least two solutions. The first is obtained explicitly by a direct calculation and the second via the Mountain Pass Theorem for the case 0 < µ 1 ≤ µ 2 < λ 1 or Linking Theorem if λ k < µ 1 ≤ µ 2 < λ k+1 , where µ 1 , µ 2 are eigenvalues of symmetric matrix A and λ j are eigenvalues of (−∆, H 1 0 (Ω)).2000 Mathematics Subject Classification. 35J50, 35B33. Key words and phrases. Ambrosetti-Prodi type problems, systems of elliptic equations, critical Sobolev exponents.

In this paper, we deal with a quasilinear elliptic system in exterior domains with dependence on the gradient and coupling of the equations not only inside of the domain, but also on the boundary. We prove the existence of positive, negative or sign changing weak solutions. Our approach relies on an aproximation argument and an adequate elliptic "a priori" estimate.