On Some Elliptic Equations With W0 1,1(Ω) Solutions
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Date
2025-06-15
Journal Title
Journal ISSN
Volume Title
Publisher
Mohamed Boudiaf University of M'sila
Abstract
This work investigates the regularizing effects of lower-order terms in nonlinear Dirichlet prob lems of the form:
(
−
u
div
= 0, on
¡
|∇u|
p−2∇u
¢ + H(x,u,∇u) = f (x), in Ω
∂
,
Ω,
(1)
where Ω ⊂ R
N (N ≥ 2) is a bounded domain, 1 < p ≤ N, and f has poor summability. We demon strate how lower-order terms can enhance solution regularity, particularly when f ∈ L
1
(Ω) or other
Lebesgue spaces.
According to the work [8], this study focuses on four principal cases:
(A) For H(x,u,∇u) = u|u|
r−2
, we establish existence of weak solutions in W0
1,2(Ω) even when
f ∈ L
1
(Ω)
(B) With polynomial nonlinearities, we prove existence of distributional solutions in W0
1,1(Ω) for
f ∈ L
r
′
/p
(Ω) (1 < p ≤ r
′
)
(C) For gradient-dependent terms H(x,u,∇u) = u|u|
r−2
|∇u|, we obtain solutions in W0
1,1(Ω) ∩
L
r−1
(Ω) when f ∈ L
1
(Ω) and 1 < r ≤ N
N
(p
−
−
1
1)
(D) We compare these results with the semilinear case (p = 2), highlighting differences in regu larization mechanisms
The analysis employs a unified three-step approach: (1) approximation by regular problems, (2)
derivation of a priori estimates in W0
1,1(Ω), and (3) passage to the limit. Our results significantly ex tend previous work by demonstrating existence in borderline cases where the unperturbed prob lem (H = 0) admits no solutions. The findings have important implications for understanding
nonlinear elliptic equations with non-regular data.
Description
Keywords
Nonlinear Dirichlet problem, Existence, Regularity, Regularizing effects, Non-regular data, W0 1, 1(Ω)