Qualitative properties of positive solutions to mixed local and nonlocal critical problems in ${\mathbb{R}}^n$

We consider the following mixed local and non-local critical elliptic equation: \begin{equation*}\label{0.1} \left{ \begin{array}{lll} -Δu+(-Δ)^su=λh u^{p}+u^{2^-1}, &$\text{in}$,, $\mathbb{R}$^n, u>0, &\text {in} ,, $\mathbb{R}$^n, \lim\limits_{|x|\to\infty} u(x) = 0, \end{array} \right. \end{equation} where $n⩾4,p\in (0,2^{\ast }-1),2^{\ast }:=\frac{2n}{n-2}$ and $h$ is a positive function. We first show the existence and regularity results of viscosity solutions to the above critical elliptic equation. More precisely, from \cite{Su-Xu} weak solutions are obtained and we prove they are indeed viscosity solutions and their regularity is: $u\in C^{\alpha }({\mathbb{R}}^n)$ for $p\in (0,1);$ $u\in C^{2,\beta }({\mathbb{R}}^n)$ for $p\in [1,2^{\ast }-1)\mathrm{.}$ Moreover, for $p\in [1,2^{\ast }-1)$, these viscosity solutions are indeed classical ones and we then prove the existence of positive solutions with the qualitative properties such as the decay estimates and the radial symmetry.