Minkowski空间中含一类推广的平均曲率算子系统的正径向解存在唯一性

Existence and Uniqueness of Positive Radial Solutions for a Class of Generalized Mean Curvature Operator Systems in Minkowski Space

摘要: 本文运用锥上不动点指数理论探讨含一类推广的平均曲率算子的拟线性微分系统Dirichlet问题$\left(H\right)\{\begin{array}{l}{M}_k\left(u\right)+f_1\left(v\right)=0\text{, }x\in {ℬ}_1\\ M_k\left(v\right)+f_2\left(u\right)=0\text{, }x\in {ℬ}_1\\ {u|}_{\partial {ℬ}_1}={v|}_{\partial {ℬ}_1}=0\end{array}$ 径向正解的存在性和唯一性，其中$M_k\left(u\right)=div\left(\frac{\nabla u}{{\left(1-{\left|\nabla u\right|}^2\right)}^{\frac{k}{2}}}\right)\left(k\ge 1\right)$ ，$f_i\in C\left(\left[0,\infty \right),\left[0,\infty \right)\right),i=1,2$ ，${ℬ}_1$ 为${ℝ}^N\left(N\ge 2\right)$ 空间中的单位球。当$k=1$ 时，$M_1\left(u\right)$ 即为经典的平均曲率算子。

Abstract: This paper employs the fixed point index theory in cones to investigate the existence and uniqueness of positive radial solutions for the Dirichlet problem of the following quasilinear differential system:$\left(H\right)\{\begin{array}{l}{M}_k\left(u\right)+f_1\left(v\right)=0\text{, }x\in {ℬ}_1\\ M_k\left(v\right)+f_2\left(u\right)=0\text{, }x\in {ℬ}_1\\ {u|}_{\partial {ℬ}_1}={v|}_{\partial {ℬ}_1}=0\end{array}$ .Here, $M_k\left(u\right)=div\left(\frac{\nabla u}{{\left(1-{\left|\nabla u\right|}^2\right)}^{\frac{k}{2}}}\right)\left(k\ge 1\right)$ , $f_i\in C\left(\left[0,\infty \right),\left[0,\infty \right)\right),i=1,2$ , and ${ℬ}_1$ denotes the unit ball in ${ℝ}^N\left(N\ge 2\right)$ . When $k=1$ , $M_1\left(u\right)$ corresponds to the classical mean curvature operator.