Liouville-type theorems for Lane--Emden inequalities involving nonlocal operators
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912904508342272 |
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| author | Kim, T. Lee, T. |
| author_facet | Kim, T. Lee, T. |
| contents | We establish a Liouville-type theorem for nonnegative weak supersolutions to $\mathcal{L}_K u = u^q$ in $\mathbb{R}^n$, where $\mathcal{L}_K$ is a translation-invariant integro-differential operator of order $2s$ with $s \in (0,1)$. The kernel $K$ is assumed to be even and satisfy uniform ellipticity bounds. We prove that the only nonnegative supersolution is the trivial one $u \equiv 0$ in the range $1 < q \le \frac{n}{n-2s}$ for $n > 2s$ (and for all $q > 1$ when $n \le 2s$). Our proof is elementary and relies on a test function method combined with a dyadic decomposition of the nonlocal tail. Notably, our argument does not rely on the maximum principle or the fundamental solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_13822 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Liouville-type theorems for Lane--Emden inequalities involving nonlocal operators Kim, T. Lee, T. Analysis of PDEs 35B53 (Primary) 35R11, 35J61 (Secondary) We establish a Liouville-type theorem for nonnegative weak supersolutions to $\mathcal{L}_K u = u^q$ in $\mathbb{R}^n$, where $\mathcal{L}_K$ is a translation-invariant integro-differential operator of order $2s$ with $s \in (0,1)$. The kernel $K$ is assumed to be even and satisfy uniform ellipticity bounds. We prove that the only nonnegative supersolution is the trivial one $u \equiv 0$ in the range $1 < q \le \frac{n}{n-2s}$ for $n > 2s$ (and for all $q > 1$ when $n \le 2s$). Our proof is elementary and relies on a test function method combined with a dyadic decomposition of the nonlocal tail. Notably, our argument does not rely on the maximum principle or the fundamental solution. |
| title | Liouville-type theorems for Lane--Emden inequalities involving nonlocal operators |
| topic | Analysis of PDEs 35B53 (Primary) 35R11, 35J61 (Secondary) |
| url | https://arxiv.org/abs/2602.13822 |