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

Abstract

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.