Mixed fully nonlinear local and nonlocal elliptic operators in Heisenberg group

Abstract

We establish the comparison principle, existence and regularity of viscosity solutions to the following problem concerning the mixed operator: \begin{align} \begin{cases} \alpha\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}u\big)-\beta\big(-\Delta_{\mathbbm{H}^N}\big)^su=f &\text{in } \,{\Omega}, u=g &\text{in } \,\mathbbm{H}^N\setminus\Omega, \end{cases} \end{align} where $\mathcal{M}_{\lambda,\Lambda}^+$ is the extremal Pucci's operator and $(-\Delta_{\mathbbm{H}^N})^s$ denotes the fractional sub-Laplacian on Heisenberg group. Here $\alpha\geq 0$ and $\beta>0$ are constants.