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Comparison principle and Liouville type results for singular fully nonlinear operators. (English) Zbl 1129.35369
Summary: We consider a large class of degenerate or singular operators $F$ defined on $ℝ×{\left({ℝ}^{N}\right)}^{*}×S$, where $S$ denotes the space of symmetric matrices on ${ℝ}^{n}$, $F$ is continuous. We give a new definition of viscosity sub and super solutions for $F\left(x,\nabla u,{\nabla }^{2}u\right)=0$. We prove a comparison theorem between sup and supersolutions for $F\left(x,\nabla u,{\nabla }^{2}u\right)=b\left(u\right)$ where $b$ is an increasing function, and a Liouville type result.
##### MSC:
 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE) 35B50 Maximum principles (PDE)