  In this paper we study double phase problems with nonlinear boundary
condition and gradient dependence. Under quite general assumptions we prove
existence results for such problems where the perturbations satisfy a suitable
behavior in the origin and at infinity. Our proofs make use of variational
tools, truncation techniques and comparison methods. The existence of the
obtained solutions depends on the first eigenvalues of the Robin and Steklov
eigenvalue problems for the $p$-Laplacian.
