We study regularity for solutions of quasilinear elliptic equations of the form $\mathrm {div}\mathbf{A}(x,u,\nabla u)=\mathrm {div}\mathbf{F}$ in bounded domains in $\mathbb{R}^n$. The vector field $\mathbf{A}$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under a small BMO condition in $x$ for $\mathbf{A}$. As a consequence, we obtain that $\nabla u$ is in the classical Morrey space $\mathcal{M}^{q,\lambda}$ or weighted space $L^q_w$ whenever $|\mathbf{F}|^{1/(p-1)}$ is respectively in $\mathcal{M}^{q,\lambda}$ or $L^q_w$, where $q$ is any number greater than $p$ and $w$ is any weight in the Muckenhoupt class $A_{q/p}$. In addition, our two-weight estimate allows the possibility to acquire the regularity for $\nabla u$ in a weighted Morrey space that is different from the functional space that the data $|\mathbf{F}|^{1/(p-1)}$ belongs to.

中文翻译：

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拟线性椭圆方程在加权Morrey空间中的正则估计

我们研究$ \的有限域中$ \ mathrm {div} \ mathbf {A}（x，u，\ nabla u）= \ mathrm {div} \ mathbf {F} $形式的拟线性椭圆方程的解的正则性mathbb {R} ^ n $。向量字段$ \ mathbf {A} $假定在$ u $中是连续的，并且其在$ \ nabla u $中的增长类似于$ p $ -Laplace运算符。我们在较小的BMO条件下，在$ x $中，对于$ \ mathbf {A} $，在方程的弱解$ u $的加权Morrey空间中建立内部梯度估计。结果，我们获得了$ \ nabla u $在经典Morrey空间$ \ mathcal {M} ^ {q，\ lambda} $或加权空间$ L ^ q_w $的情况，只要$ | \ mathbf {F} | ^ {1 /（p-1）} $分别位于$ \ mathcal {M} ^ {q，\ lambda} $或$ L ^ q_w $中，其中$ q $是大于$ p $和$ w $的任意数字是Muckenhoupt类$ A_ {q / p} $中的任何权重。此外，