A new class of double phase variable exponent problems: Existence and uniqueness

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 corresponding Musielak-Orlicz Sobolev spaces (an equivalent norm, uniform convexity, Radon-Riesz property respect to modular) and (continuity, strict monotonicity, (S+)-property). contrast known constant exponent case are able weaken assumptions on data. Finally show existence uniqueness right-hand sides that have gradient dependence (so-called convection terms) under very general As result independent interest, also density smooth functions in space even when domain is unbounded.