Anisotropic equations in weighted Sobolev spaces of higher order

Abstract

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.

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

“…On the other hand, using duality arguments, we deduce that L 1 ( ) ⊂ W −(n+1), − → p ( , − → ) and the case f ∈ L 1 ( ) may be considered as dual case (see [10]). …”

“…Moreover from assumptions (A 1 ), (A 2 ) and (A 3 ), we deduce that A 2n+2 satisfies the growth, the coerciveness and the monotonicity conditions. Thanks to ( [10], Theorem 3.1), it follows from the theory of pseudo-monotone operators that there exists at least one solution u n ∈ W n+1, − → p 0 ( , − → ) of the following approximate problem:…”

“…Finally, it would be interesting at this stage to refer the reader to the work of Dubinskiȋ [9] where the author proved, under hypothesis (A 1 )−(A 4 ) (see below), the existence of solution for the Dirichlet problem associated with the equation Au = f and to the work [10] dealing with problems of finite and infinite order of type (1) in the setting of weighted anisotropic Sobolev spaces with data in dual.…”

Anisotropic Elliptic Nonlinear Obstacle Problem with Weighted Variable Exponent

“…The study of (P ) is a new and interesting topic when the data is in L 1 . One result on this topic can be found in [5,8,11], where the discussion was conducted in the framework of weighted anisotropic Sobolev space with variable exponent (we refer to [1,2,11] for more details), the notion of a entropy solution was introduced by Benilan et. al [7,9] and P.-L. Lions [14] in their study of the Boltzmann equation.…”

“…The aim of this paper is to extend the results in [5] to the anisotropic obstacle nonlinear elliptic problem. We want to prove only existence results, the uniqueness problem being a rather delicate one, this kind of problems still attracting the interest of the researchers (see [10,11,15] for a survey). One of the motivations for studying (P ) comes from applications to elasticity as the equations that models the shape of an elastic membrane which is pushed by an obstacle from one side affecting its shape.…”