Liouville results for semilinear integral equations with conical diffusion
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| Format: | Preprint |
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2024
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| _version_ | 1866917950440603648 |
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| author | Birindelli, Isabeau Du, Lele Galise, Giulio |
| author_facet | Birindelli, Isabeau Du, Lele Galise, Giulio |
| contents | Nonexistence results for positive supersolutions of the equation $$-Lu=u^p\quad\text{in $\mathbb R^N_+$}$$ are obtained, $-L$ being any symmetric and stable linear operator, positively homogeneous of degree $2s$, $s\in(0,1)$, whose spectral measure is absolutely continuous and positive only in a relative open set of the unit sphere of $\mathbb R^N$. The results are sharp: $u\equiv 0$ is the only nonnegative supersolution in the subcritical regime $1\leq p\leq\frac{N+s}{N-s}\,$, while nontrivial supersolutions exist, at least for some specific $-L$, as soon as $p>\frac{N+s}{N-s}$. \\ The arguments used rely on a rescaled test function's method, suitably adapted to such nonlocal setting with weak diffusion; they are quite general and also employed to obtain Liouville type results in the whole space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_12720 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Liouville results for semilinear integral equations with conical diffusion Birindelli, Isabeau Du, Lele Galise, Giulio Analysis of PDEs Nonexistence results for positive supersolutions of the equation $$-Lu=u^p\quad\text{in $\mathbb R^N_+$}$$ are obtained, $-L$ being any symmetric and stable linear operator, positively homogeneous of degree $2s$, $s\in(0,1)$, whose spectral measure is absolutely continuous and positive only in a relative open set of the unit sphere of $\mathbb R^N$. The results are sharp: $u\equiv 0$ is the only nonnegative supersolution in the subcritical regime $1\leq p\leq\frac{N+s}{N-s}\,$, while nontrivial supersolutions exist, at least for some specific $-L$, as soon as $p>\frac{N+s}{N-s}$. \\ The arguments used rely on a rescaled test function's method, suitably adapted to such nonlocal setting with weak diffusion; they are quite general and also employed to obtain Liouville type results in the whole space. |
| title | Liouville results for semilinear integral equations with conical diffusion |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.12720 |