  In this paper we introduce a new class of quasilinear elliptic equations
driven by the so-called double phase operator with variable exponents. We prove
certain properties of the corresponding Musielak-Orlicz Sobolev spaces
(completeness, reflexivity, an equivalent norm) and the properties of the new
double phase operator (continuity, strict monotonicity, (S$_+$)-property).
Finally we show the existence and uniqueness of corresponding elliptic
equations with right-hand sides that have gradient dependence (so-called
convection terms) under very general assumptions on the data. As a result of
independent interest, we also show the density of smooth functions in the new
Musielak-Orlicz Sobolev space even when the domain is unbounded.
