  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 (an
equivalent norm, uniform convexity, Radon-Riesz property with respect to the
modular) and the properties of the new double phase operator (continuity,
strict monotonicity, (S$_+$)-property). In contrast to the known constant
exponent case we are able to weaken the assumptions on the data. 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.
