We establish existence and uniqueness of positive viscosity solutions of $\left\{\begin{array}{c}{\mathcal{P}}_k^{\pm }\left(D^{2}u\right)+{\left|Du\right|}^q{u}^p=0\text{in}\Omega ,\hfill \\ u=0\text{on}\partial \Omega ,\hfill \end{array}\right.$where $k<N,\Omega$ is a bounded domain in ${\mathbb{R}}^N,N\ge 2,0<p<1,0\le q<1$ and ${\mathcal{P}}_k^{\pm }$ are degenerate elliptic operators. First, we use of a change of dependent variable originating in Brezis and Kamin (1992) in order to convert the equation into one with the right monotonicity in the $u$-variable. Thereafter by applying Perron’s method, we prove the existence and uniqueness of the solutions. Using an a-priori estimate, we show the nonexistence of subsolutions. We also find the ranges of $p$ and $q$ for the existence and nonexistence results.

我们确定正粘度溶液的存在性和唯一性 $\left\{\begin{array}{c}{\mathcal{P}}_{ķ}^{\pm }（d^2ü）+{\left|dü\right|}^q{ü}^p=0\text{在}\Omega ，\hfill \\ ü=0\text{上}\partial \Omega ，\hfill \end{array}\right.$哪里 $ķ<ñ，\Omega$ 是一个有界域 ${\mathbb{[R}}^{ñ}，ñ\ge 2，0<p<\mathrm{1个}，0\le q<\mathrm{1个}$ 和 ${\mathcal{P}}_{ķ}^{\pm }$是简并的椭圆算子。首先，我们使用源自Brezis和Kamin（1992）的因变量变化来将方程转换为在方程中具有正确单调性的方程。$ü$-变量。此后，通过应用Perron方法，我们证明了解的存在性和唯一性。使用先验估计，我们显示了子解决方案的不存在。我们还找到了$p$ 和 $q$ 对于存在和不存在的结果。