Degenerate elliptic equations with lower-order terms and L 1 data
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Date
2025-06-15
Journal Title
Journal ISSN
Volume Title
Publisher
Mohamed Boudiaf University of M'sila
Abstract
This thesis investigates the existence of weak solutions for a class of degenerate elliptic
equations with lower-order terms and right-hand side data in L
1
(Ω). The problem under
consideration is of the form:
−div M
1 +
(x)
|
∇
u|
u
!
+ g(x)u = f(x) in Ω,
u = 0 on ∂Ω,
where Ω ⊂ R
N is a bounded open domain, M(x) is a bounded and elliptic matrix, g(x) ∈
L
1
(Ω) is a nonnegative lower-order coefficient, and f(x) ∈ L
1
(Ω) satisfies a domination
condition of the type |f(x)| ≤ kg(x).
Due to the lack of coercivity and low regularity of the data, we introduce a sequence of
approximate problems using truncation functions to regularize the nonlinear operator. We
then establish uniform a priori estimates for the approximate solutions in H0
1
(Ω) ∩ L
∞(Ω).
Finally, we pass to the limit in the nonlinear terms using compactness and weak convergence
techniques to prove the existence of a bounded weak solution to the original problem.
Description
Keywords
degenerate elliptic equations, lower-order terms, pseudo-monotone operator, L 1 -data, bounded weak solution