We investigate symmetry properties of positive solutions for fully nonlinear uniformly elliptic systems, such as $$ F_i \,(x,Du_i,D^2u_i) +f_i \,(x,u_1, \ldots , u_n,Du_i)=0, \;\; 1 \leq i \leq n, $$ in a bounded domain $\Omega$ in $\mathbb{R}^N$ with Dirichlet boundary condition $u_1=\ldots,u_n=0$ on $\partial\Omega$. Here, $f_i $'s are nonincreasing with the radius $r=|x|$, and satisfy a cooperativity assumption. In addition, each $f_i $ is the sum of a locally Lipschitz with a nondecreasing function in the variable $u_i$, and may have superlinear gradient growth. We show that symmetry occurs for systems with nondifferentiable $f_i$'s by developing a unified treatment of the classical moving planes method in the spirit of Gidas-Ni-Nirenberg. We also present different applications of our results, including uniqueness of positive solutions for Lane-Emden systems in the subcritical case in a ball, and symmetry for a class of systems with natural growth in the gradient.

中文翻译：

---

全非线性椭圆系统正解的对称性

我们研究了完全非线性一致椭圆系统正解的对称性质，例如 $$ F_i \,(x,Du_i,D^2u_i) +f_i \,(x,u_1, \ldots , u_n,Du_i)=0, \ ;\; 1 \leq i \leq n, $$ 在 $\mathbb{R}^N$ 中的有界域 $\Omega$ 中，在 $\partial\Omega$ 上使用 Dirichlet 边界条件 $u_1=\ldots,u_n=0$。这里，$f_i $'s 不随半径 $r=|x|$ 增加，并且满足协同性假设。此外，每个 $f_i $ 是变量 $u_i$ 中具有非递减函数的局部 Lipschitz 的总和，并且可能具有超线性梯度增长。我们通过本着 Gidas-Ni-Nirenberg 的精神开发经典移动平面方法的统一处理，表明具有不可微的 $f_i$ 的系统会出现对称性。我们还展示了我们结果的不同应用，