Abstract
We investigate a general nonlinear elliptic and parabolic equation with $L^1$-data in the anisotropic Musielak-Orlicz space avoiding growth restrictions. The growth of the monotone vector field is controlled by a generalized nonhomogeneous and anisotropic N-function. We do not assume any particular type of growth condition of M or its conjugate M^* and therefore the spaces we deal with are not reflexive.
The main results are existence and uniqueness of renormalized solutions to the above general elliptic and parabolic equations. As a main tool we provide density of smooth functions in modular topology. The condition we impose is certain type of regularity of $M(x,\xi)$ capturing interplay between behavior of $M$ for big $|\xi|$ and small changes of and space variables. Retrieving the known optimal results we exclude the Lavrentiev phenomenon in the variable exponent spaces under asymptotical log-H\"older continuity assumption and in the double-phase space within the sharp range of parameters.
In order to get existence, the regularity assumption can be simply skipped not only in the Orlicz case ($M(x,\xi)=M(\xi)$), but also in reflexive spaces (e.g. if $M,M^*\in\Delta_2$), that is among others in the variable exponent, weighted Sobolev and the double phase space, no matter how irregular the exponent or the weights are.
