In this work, generalized Sobolev spaces and Sobolev spaces of infinite order are considered. Existence of solutions for strongly nonlinear equation of infinite order of the form Au + g(x, u) = f is established. Here A is an elliptic operator from a functional space of Sobolev type to its dual and g(x, s) is a lower order term satisfying a sign condition on s.

We consider the strongly nonlinear boundary value problem,where A is an elliptic operator of finite or infinite order. We introduce anisotropic weighted Sobolev spaces and we show under a certain sign condition of the Carathéodory function g without assuming any growth restrictions, the existence of the weak solutions.

In this paper, we give an approximation result in some anisotropic Sobolev space. We also describe the action of some distributions in the dual and we mention two applications to some strongly nonlinear anisotropic elliptic boundary value problems.

In this article, we shall be concerned with the existence of solutions for the strongly non-linear boundary value problem:where A is an elliptic operator of finite order defined from an anisotropic Sobolev space of order m to its dual, g is a Carathe´odory function satisfying essentially a sign condition on u with no growth restrictions and f belongs to L 1 .

We deal with the existence and uniqueness of weak solutions for a class of strongly nonlinear boundary value problems of higher order with L 1 data in anisotropic-weighted Sobolev spaces of infinite order.