On renormalized solutions to elliptic inclusions with nonstandard growth

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