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A general boundary value problem of variational type is considered for a general quasi-linear elliptic partial differential operator of order 2m in generalized divergence form. Such problems are considered on an arbitrary domain in a Euclidean space without hypotheses of boundedness on the domain or smoothness on its boundary. Contrary to the prevailing doctrine in the literature, it is shown that the corresponding operator between Banach spaces is pseudo-monotone, and that a wide variety of existence results can be derived from this fact.

Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains