Nonlinear Weighted Elliptic Problem With Degenerate Coercivity And L 1 Data

Résumé: 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).

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