| Abstract: |
| We establish Liouville-type theorems for nonnegative weak supersolutions of nonlocal Lane--Emden equations $L_K u = u^q$ in $\mathbb{R}^n$. For linear integro-differential operators of order $2s$, $s\in(0,1)$, we prove that $u\equiv 0$ when $1\lt q\leq n/(n-2s)$, via a test function method combined with a dyadic decomposition of the nonlocal tail. We then extend the analysis to quasilinear operators of $p$-Laplacian type associated with general kernels satisfying mild structural conditions, where a singular test function $u^{-\alpha}\varphi$ yields the corresponding nonexistence result. The arguments are elementary, relying only on the weak formulation rather than the maximum principle, fundamental solutions, or extension methods, and are motivated by blow-up analysis in regularity theory. |
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