On a strongly anisotropic equation withL¹data

Abstract

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 .

Elliptic equations in weighted Sobolev spaces of infinite order with L 1 data

“…[4--7]). Let us also mention that in the recent work [8], the authors proved the existence of solutions in the setting of anisotropic spaces of finite order. It is the purpose of this work to get the existence results in the setting of anisotropic-weighted Sobolev spaces for a class of nonlinear elliptic equations of infinite order of type (1) with L 1 data, which include as a special case problems involving Leray-Lions operators in the usual sense.…”

Strongly anisotropic elliptic problems with regular and $L^1$ data

“…Let us point out that an interesting work in this direction can be found in [17] where the authors proved the existence of renormalized solutions for some anisotropic quasilinear elliptic equations. Finally, it would be interesting to mention that when g does not depend on Du; we refer the reader to the works [3] and [8], dealing with strongly nonlinear elliptic equations governed by a general class of anisotropic operators.…”

Some results on strongly nonlinear anisotropic differential equations

“…in [7], [10] and [14]. Finally, when g does not depend on ∇u, we refer the reader to the recent works [3] and [8], dealing with elliptic equations for a general class of operators of finite and infinite order, and proving the existence of solutions in anisotropic spaces.…”