  We study a general class of quasilinear elliptic equations with nonstandard
growth to prove the existence of a very weak solution to such a problem. A key
ingredient in the proof is a priori global weighted gradient estimate of a very
weak solution, where the right hand side of the equation is the divergence of a
vector-valued function with low degree of integrability. To obtain this
estimate, we adopt a notion of reverse H\"older class of Muckenhoupt weights.
Another crucial part of the proof is a generalized weighted div-curl lemma in
the setting of Orlicz spaces.
