On Some Elliptic Equations With W0 1,1(Ω) Solutions

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.