  In this paper we study quasilinear elliptic equations driven by the so-called
double phase operator and with a nonlinear boundary condition. Due to the lack
of regularity, we prove the existence of multiple solutions by applying the
Nehari manifold method along with truncation and comparison techniques and
critical point theory. In addition, we can also determine the sign of the
solutions (one positive, one negative, one nodal). Moreover, as a result of
independent interest, we prove for a general class of such problems the
boundedness of weak solutions.
