Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients

Variational boundary value problems for quasilinear elliptic systems in divergence form are studied in the case where the nonlinearities are nonpolynomial. Monotonicity methods are used to derive several existence theorems which generalize the basic results of Browder and Leray-Lions. Some features of the mappings of monotone type which arise here are that they act in nonreflexive Banach spaces, that they are unbounded and not everywhere defined, and that their inverse is also unbounded and not everywhere defined.