We consider the following mixed local and non-local critical elliptic equation: \begin{ equation*}\label{ 0.1} \left\{ \begin{ array}{ lll} -\Delta u+(-\Delta)^su=\lambda h 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 $n\geqslant4, \,\, p\in (0,2^*-1),\,\, 2^*:=\frac{ 2n}{ n-2}$ 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 $p\in [1, 2^*-1).$ Moreover, for $p\in [1, 2^*-1)$, 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.
Categories: math.AP