Existence of solutions for a class of obstacle problems with $$L^1$$ L 1 -data and without sign condition

Abstract

In this article, we prove an existence result of solutions to the obstacle problem associated with the equation of the typewhere A is an operator of Leray-Lions type acting from W 1, p(x) 0 ( ) into its dual W −1, p (x) ( ) and g(x, s, ξ) is a nonlinear term satisfying some growth condition,without the sign condition.

On some $\overrightarrow{p(x)}$ anisotropic elliptic equations in unbounded domain

“…In this section we present the anisotropic variable exponent Sobolev space, used in the study of the elliptic problem (1). Let p 0 (x), p 1 (x), ..., p N (x) be N + 1 variable exponents in C + (Ω).…”

“…), ..., p N (.)) where p i : Ω → IR are measurable functions, an excellent introduction is in ( [18]), another sources are ( [1], [5], [11], [17] [25]).…”

The obstacle problem for degenerate anisotropic elliptic equations with variable exponents and $L^1$-data

“…. , N, the existence of entropy solutions to the obstacle problems associated with the operator A has been widely investigated in literature (see, for example, [6,7] and references therein).…”

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).…”