Convergence of weak solutions of elliptic problems with datum in L 1

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 { − Δ p u ε + ε | f ( x ) | u ε = f ( x ) in  Ω , u ε = 0 on  ∂ Ω , when ε tends to 0 . Here, Ω denotes a bounded open set of R N ( N ≥ 2 ) , − Δ p u = − d i v ( | ∇ u | p − 2 ∇ u ) is the usual p -Laplacian operator ( 1 < p < ∞ ) and f ( x ) is an L 1 ( Ω ) function. We show that this sequence converges in some sense to u , the entropy solution of the problem { − Δ p u = f ( x ) in  Ω , u = 0 on  ∂ Ω . In the semilinear case, we prove stronger results provided the weak solution of that problem exists.