Global boundedness of weak solutions with finite energy to a general class of Dirichlet problems

As explained in detail in the prologue to this manuscript, boundedness of weak solutions for general classes of elliptic equations in divergence form is a classic tool for achieving higher regularity. We propose here some global boundedness results under general assumptions that can be applied to several cases studied in the recent and extensive literature on partial differential equations under general growth. In particular, we propose the class of weak solutions with finite energy in which to search for solutions and in which regularity can be studied and achieved. We emphasize that we are not limited to minimizers of certain integral functionals, as often considered recently in this context of general growth, but to the broader class of weak solutions to Dirichlet problems for general nonlinear elliptic equations in divergence form. When applied to p, q problems, a further major novelty is that we consider general p, q growth conditions for the principal part of the differential equation, but we do not need an upper bound on the ratio q p , we just need nothing more than 1 < p <= q .