A pointwise gradient bound for weak solutions to Dirichlet problem for quasilinear elliptic equations $ -\mathrm{div}(\mathbb{A}(x,\nabla u)) = \mu $ is established via Wolff type potentials. It is worthwhile to note that the model case of $ \mathbb{A} $ here is the non-degenerate $ p $-Laplacian operator. The central objective is to extend the pointwise regularity results in [Q.-H. Nguyen, N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equations with measure data, J. Funct. Anal. 278(5) (2020), 108391] to the very singular case $ 1<p \le \frac{3n-2}{2n-1} $, where the data $ \mu $ on right-hand side is assumed belonging to some classes that close to $ L^1 $. Moreover, a global pointwise estimate for gradient of weak solutions to such problem is also obtained under the additional assumption that $ \Omega $ is sufficiently flat in the Reifenberg sense.

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一类非常奇异的拟线性椭圆型方程的逐点梯度界

通过Wolff型势建立了拟线性椭圆型方程Dirichlet问题的弱解的逐点梯度界--mathrm {div}（\ mathbb {A}（x，\ nabla u））= \ mu $。值得注意的是，这里的\ \ mathbb {A} $的模型情况是非退化$ p $ -Laplacian运算符。中心目标是扩展[Q.-H.]中的逐点规则性结果。Nguyen，NC Phuc，一类具有测量数据的奇异拟线性方程的点向梯度估计，J。Funct。肛门 278（5）（2020），108391]将很奇异情况$ 1 <p \ le \ frac {3n-2} {2n-1} $，其中右侧的数据$ \ mu $被认为属于某些接近$ L ^ 1 $的类。而且，在另外的假设下，在Reifenberg的意义上，ω\Ω是足够平坦的，还获得了针对该问题的弱解的梯度的全局逐点估计。