Liouville Theorems for the Lane-Emden Equation Involving a Mixed Local-Nonlocal Operator

In this paper, we investigate the existence of positive supersolutions for the following mixed local-nonlocal Lane-Emden type equation: $$-\mathrm{\Delta }u+(-\mathrm{\Delta }{)}^{s}u=u^q\text{in }{\mathbb{R}}^n,$$ where $n\ge 3$, $s\in (0,1)$, and $q>1$. More precisely, we prove that the equation admits positive distributional supersolutions if and only if $q>\frac{n}{n-2s}$. In the process, we establish several novel properties of mixed local-nonlocal operators, including sharp asymptotic estimates for the fundamental solution, a maximum principle in the distributional sense, and an equivalent integral inequality for supersolutions.