On renormalized solutions to elliptic inclusions with nonstandard growth

We study the elliptic inclusion given in following divergence form $$\begin{aligned}&-\mathrm {div}\,A(x,\nabla u) \ni f\quad \mathrm {in}\quad \Omega ,\\&u=0\quad {on}\quad \partial . \end{aligned}$$ As we assume that $$f\in L^1(\Omega )$$ , solutions to above problem are understood renormalized sense. also nonstandard, possibly nonpolynomial, heterogeneous and anisotropic growth coercivity conditions on maximally monotone multifunction A which necessitates use of nonseparable nonreflexive Musielak–Orlicz spaces. prove existence uniqueness solution as well as, under additional assumptions data, its boundedness. The key difficulty, lack a Carathéodory selection is overcome with Minty transform.