We solve the stationary Navier-Stokes equations for non-Newtonian incompressible fluids with shear dependent viscosity in domains with unbounded outlets, in the case of shear thickening viscosity, i.e. the viscosity is given by the shear rate raised to the power p- 2 where p> 2. The flux assumes arbitrary given values and the Dirichlet integral of the velocity field grows at most linearly in the outlets of the domain. Under some smallness conditions on the "energy dispersion" we also show that the solution of this problem is unique. Our results are an extension of those obtained by O.A. Ladyzhenskaya and V.A. Solonnikov [O.A. Ladyzhenskaya, V.A. Solonnikov, Determination of the solutions of boundary value problems for steady-state Stokes and Navier-Stokes equations in domains having an unbounded Dirichlet integral, J. Soviet Math. 21 (1983) 728-761] for Newtonian fluids (p= 2). © 2011 Elsevier Inc.252638733898Amick, C.J., Steady solutions of the Navier-Stokes equations in unbounded channels and pipes (1977) Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 4 (3), pp. 473-513Barrett, J.W., Liu, W.B., Finite element approximation of the p-Laplacian (1993) Math. Comp., 61 (204), pp. 523-537Beirão da Veiga, H., Kaplický, H., Růžička, M., Boundary regularity of shear thickening flows (2011) J. Math. Fluid Mech., 13, pp. 387-404Bird, R., Stewart, W., Lightfoof, E., (2007) Transport Phenomena, , Johh Wiley & Sons, IncDiBenedetto, E., (1994) Degenerate Parabolic Equations, , Springer-Verlag, BerlinEvans, L.C., Partial Differential Equations (1998) Grad. Stud. Math., 19. , Amer. Math. Soc., Providence, RIFrehse, J., Málek, J., Steinhauer, M., On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method (2003) SIAM J. Math. Anal., 34 (5), pp. 1064-1083Galdi, G.P., (1994) An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vols. I, II, , Springer-Verlag, BerlinLadyzhenskaya, O.A., New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them (1967) Proc. Steklov Inst. Math., 102, pp. 95-118Ladyzhenskaya, O.A., On some modifications of the Navier-Stokes equations for large gradients of velocity (1968) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7, pp. 126-154Ladyzhenskaya, O.A., (1969) The Mathematical Theory of Viscous Incompressible Flow, , Gordon and Breach Science Publishers, New York, London, ParisLadyzhenskaya, O.A., Solonnikov, V.A., Determination of the solutions of boundary value problems for steady-state Stokes and Navier-Stokes equations in domains having an unbounded Dirichlet integral (1983) Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov. (LOMI). J. Soviet Math., 21, pp. 728-761. , English transl.:Lions, J.L., (1969) Quelques Méthods de Resolution des Problémes Aux Limites Non Linéaires, , Dunod, Gauthier-VillarsMarušić-Paloka, E., Steady flow of a non-Newtonian fluid in unbounded channels and pipes (2000) Math. Models Methods Appl. Sci., 10 (9), pp. 1425-1445Neff, P., On Korn's first inequality with non-constant coefficients (2002) Proc. Roy. Soc. Edinburgh Sect. A, 132 (1), pp. 221-243Passerini, A., Patria, M.C., Thater, G., Steady flow of a viscous incompressible fluid in an unbounded "funnel-shaped" domain (1997) Ann. Mat. Pura Appl. (4), 173, pp. 43-62Silva, F.V., On a lemma due to Ladyzhenskaya and Solonnikov and some applications (2006) Nonlinear Anal., 64, pp. 706-725Silva, F.V., (2004), http://cutter.unicamp.br/document/?code=vtls000316769, Os problemas de Leray e de Ladyzhenskaya-Solonnikov para fluidos micropolares (Leray and Ladyzhenskaya-Solonnikov problems for micropolar fluids), doctoral thesis, in portugueseIMECC-Unicamp-BrazilSmagorinsky, J.S., General circulation experiments with the primitive equations. I. The basic experiment (1963) Mon. Weather Rev., 91, pp. 99-164Solonnikov, V.A., Pileckas, K.I., Certain spaces of solenoidal vectors, and the solvability of a boundary value problem for a system of Navier-Stokes equations in domains with noncompact boundaries (1986) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI). J. Soviet Math., 34, pp. 2101-2111. , English transl.:Stein, E.M., (1970) Singular Integrals and Differentiability Properties of Functions, , Princeton University Press, Princeton