  In this paper we study quasilinear elliptic systems driven by variable
exponent double phase operators involving fully coupled right-hand sides and
nonlinear boundary conditions. The aim of our work is to establish an enclosure
and existence result for such systems by means of trapping regions formed by
pairs of sup- and supersolutions. Under very general assumptions on the data we
then apply our result to get infinitely many solutions. Moreover, we also
discuss the case when we have homogeneous Dirichlet boundary conditions and
present some existence results for this kind of problem.
