*(English)*Zbl 1142.35041

The authors prove some variants of the comparison principle for viscosity sub- and supersolutions of fully nonlinear elliptic equations $F(x,u,Du,{D}^{2}u)=0$ in a domain ${\Omega}\subset {\mathbb{R}}^{n}$, extending standard results. Among others, the following case is considered: at any $x\in {\Omega}$, $F$ is strictly increasing with respect to $u$ or $F$ is non-totally degenerate, what roughly means that $F(x,u,p,M+rI)$ is a strictly decreasing function of the real parameter $r$, with $I$ being the identity matrix and $M$ an arbitrary symmetric matrix. A further case refers to operators of the form

where $G$ is uniformly elliptic and $\sigma $ is an $n\times m$ matrix-valued function satisfying a non-degeneracy condition. A number of more specific equations is considered including Bellman-Isaacs equations, quasilinear subelliptic equations, and equations involving Pucci-type oerators. The results are applied to deduce existence theorems for the Dirichlet problem in the viscosity setting.

##### MSC:

35J70 | Degenerate elliptic equations |

35J25 | Second order elliptic equations, boundary value problems |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

49L25 | Viscosity solutions (infinite-dimensional problems) |

35H20 | Subelliptic PDE |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. (PDE) |