In this work, motivated by [T.F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight function, Nonlinear. Anal. 68 (2008) 1733–1745], and using recent ideas from Brown and Wu [K.J. Brown, T.F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign changing weight function, J. Math. Anal. Appl. 337 (2008) 1326–1336], we prove the existence of nontrivial nonnegative solutions to the nonlinear elliptic system
