Pointwise gradient estimates for a class of singular quasilinear equations with measure data

Pointwise gradient bounds for a class of very singular quasilinear elliptic equations

“…Following the previous results and being motivated by some intensive studies from Calderón-Zygmund estimates and global regularity results under the certain range of p, our interest here is to further establish these pointwise estimates for gradient of solutions to quasilinear elliptic equations in terms of Wolff type potentials of the source term µ, under 'very singular' case for which 1 < p ≤ n−2 2n−1 . Our main approach based on the technique presented by Nguyen and Phuc in [38] and our pointwise gradient potential estimates obtained in this paper extend such kind of results under assumptions (A1), (A2), (A3), (A4). Moreover, in order to obtain the global pointwise bounds for the gradient of solutions to problem (1) (the second main result in this paper), it is important for us to impose an additional assumption on the boundary of domain Ω.…”

“…Let us firstly recall the following technical lemma, Lemma 2.1, a result related to the comparison estimate between solutions to both problems (1) and the homogeneous one. This result was proved by Q.-H. Nguyen in his earlier unpublished work [36] and an initial version of the result has been proved in [37] when µ ∈ L 1 (Ω) and 3n−2 2n−1 < p ≤ 2 − 1 n , see also [47, Appendix A] for a detail proof. It is worth noting that the proof of this comparison result in [47] is still valid for the non-degenerate case τ > 0.…”

“…In some independent works, the singular range of p for which 1 < p ≤ 2 − 1 n started to get attention in the last years. Especially, we refer to an interesting paper by Nguyen and Phuc in [38], where they have dealt with the pointwise gradient estimates for the case where 3n−2 2n−1 < p ≤ 2 − 1 n by using the means of Wolff type potentials. Motivated by the contributions above-mentioned and by the ideas recently studied in [38], in this paper we are devoted to the study of pointwise gradient bounds for (1) when 1 < p ≤ 3n−2 2n−1 .…”

Quasilinear elliptic equations with a source reaction term involving the function and its gradient and measure data

“…be such that dist (supp (µ), ∂Ω) > 0. If for some C > 0 depending on p, N , q j and Λ j (j=1,2), Ω and dist (supp (µ), ∂Ω) there holds 16) then problem (1.4) admits a renormalized solution u satisfying 17) if p > 2, and…”

“…Theorem 3.7[10,14,17] Let Ω be a bounded domain of R N . Then there exists a constantC = C(N, p, Λ 1 , Λ 2 , diam(Ω)) > 0 such that if µ ∈ C b (Ω) and u is a solution of problem (3.1) there holds |∇u(x)| ≤ C (I r 1 [|µ|](x))for any B r (x) ⊆ Ω and for some γ 0 ∈ (0, N (p−1) N −1 ).…”