Convergence of weak solutions of elliptic problems with datum in L1

Martínez Aparicio, Antonio Jesús (2023) Convergence of weak solutions of elliptic problems with datum in L1. ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, 2023 (21). pp. 1-13. ISSN 1417-3875

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Abstract

Motivated by the Q-condition result proven by Arcoya and Boccardo in [J. Funct. Anal. 268(2015), No. 5, 1153–1166], we analyze the behaviour of the weak solutions {uε} of the problems −∆puε +ε|f(x)|uε = f(x) in Ω, uε = 0 on ∂Ω, when ε tends to 0. Here, Ω denotes a bounded open set of RN (N ≥ 2), −∆pu = −div(|∇u|p−2∇u) is the usual p-Laplacian operator (1 < p < ∞) and f(x) is an L1(Ω) function. We show that this sequence converges in some sense to u, the entropy solution of the problem −∆pu = f(x) in Ω, u =0 on ∂Ω. In the semilinear case, we prove stronger results provided the weak solution of that problem exists.

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