Quasilinear degenerate elliptic unilateral problems

Abstract

We will be concerned with the existence result of a degenerate elliptic unilateral problem of the form Au+H(x,u,∇u)=f, where A is a Leray-Lions operator from W1,p(Ω,w) into its dual. On the nonlinear lower-order term H(x,u,∇u), we assume that it is a Carathéodory function having natural growth with respect to |∇u|, but without assuming the sign condition. The right-hand side f belongs to L1(Ω)

Degenerate anisotropic variational inequalities with L1-data

“…. , n} > (n−1)q n(q− 1) , and examples show that the violation of the latter inequality may lead to the violation of the belonging of T -solutions of variational inequalities to W 1,1 0 (Ω). The mentioned inequality for the minimum of the numbers q i has already been used in [7] in the study of the solvability of nondegenerate anisotropic equations with measure data.…”

“…As far as elliptic variational inequalities with the same peculiarities are concerned, the known results are obtained mainly with the use of the same approaches and techniques which 'work' in the case of equations. These results relate, for the most part, to the unilateral Dirichlet problem (see for instance [1,4,5,10,12,13]), and in most cases (except for [10,13]) it is assumed that the corresponding sets of constraints contain bounded functions. Solvability of nonlinear variational inequalities with L 1 -right-hand sides E-mail addresses: aakovalevsky@yahoo.com (A.A. Kovalevsky), yuliya_gorban@mail.ru (Y.S.…”

“…Gorban). 1 Partly supported by INTAS Project No. 05-1000008-7921. and bilateral obstacles was studied in [2].…”

“…There it was also supposed that the corresponding set of constraints contains bounded functions. We observe that variational inequalities were considered in [4,5,10,12,13] for nondegenerate and in [1,2] for degenerate isotropic operators.…”

Existence of solutions of the Neumann problem for a class of equations involving the p-Laplacian in weighted Sobolev spaces

“…So in this case, ϕ defined by (1.1) ϕ(r 0 ) = 0, and by Theorem 1.1, for each µ > 0, Φ + µΨ has a local minimizer, in particular, when µ = 1, Φ + Ψ has a local minimizer which is a solution of (1.3). Let us point out, that another work in this space can be found in [1] and [2]. This paper is organized as follows: In Section 2, we present some necessary preliminary knowledge on weighted Lebesgue and Sobolev spaces.…”