Nonlinear weighted elliptic problem with degenerate coercivity and L 1 data
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Date
2024-06
Journal Title
Journal ISSN
Volume Title
Publisher
Mohamed Boudiaf University of M'sila
Abstract
In this work, we prove the existence of a weak solution of elliptic problem (P) defined by:
(P)
(
−div(a(x)
|∇u|
p−2∇u
1+|u|
) + e(x)|u|
p−2u = f in Ω;
u = 0 on ∂Ω,
with f ∈ L
1
(Ω) and the operator Au = −div(a(x)
|∇u|
p−2∇u
1+|u|
), 1 < p < ∞ is not coercive
on W
1,p
0
(Ω) despite being well-defined between W
1,p
0
(Ω) and its dual W−1,p0
(Ω). Degenerate
coercivity implies that as |u| becomes large, 1
1+|u|
tends to zero. To solve this issue, we are
going to approximate the operator by employing truncations in 1
1+|u|
to obtain a coercive
differential operator. Next, we will prove some a priori estimates on the sequence of approximate solutions, and we shall finally pass to the limit in the approximate problems to
establish the existence of a weak solution for the problem (P).
Description
Keywords
weighted Sobolev spaces, pseudo-monotone, operator nonlinear, elliptic equation, weak solution