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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2105.02535 |
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| _version_ | 1866913572095787008 |
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| author | Peng, Fa Zhang, Yi Ru-Ya Zhou, Yuan |
| author_facet | Peng, Fa Zhang, Yi Ru-Ya Zhou, Yuan |
| contents | Let $ 0\le f\in C^{0,1}(\mathbb R^n)$. Given a domain $Ω\subset \mathbb R^n$, we prove that any stable solution to the equation $-Δu=f(u)$ in $Ω$ satisfies a BMO interior regularity when $n=10$, and an Morrey $M^{p_n,4+2/(p_n-2)}$ interior regularity when $n\ge 11$, where $$p_n=\frac{2(n-2\sqrt{n-1}-2)}{n-2\sqrt{n-1}-4}. $$
This result is optimal as hinted by earlier results, and answers an open question raised by Cabré, Figalli, Ros-Oton and Serra. As an application, we show a sharp Liouville property: Any stable solution $u \in C^2(\mathbb R^n)$ to $-Δu=f(u)$ in $\mathbb R^n$ satisfying the growth condition, i.e.\ $|u(x)|= o\left( \log|x| \right)$ as $|x|\to+\infty$ when $n=10$; or $|u(x)|= o\left( |x| ^{ -\frac n2+\sqrt{n-1}+2 }\right)$ as $x|\to+\infty$ when $n\ge 11$, must be a constant. This extends the well-known Liouville property for radial stable solutions obtained by Villegas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_02535 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Optimal regularity & Liouville property for stable solutions to semilinear elliptic equations in $\mathbb R^n$ with $n\ge10$ Peng, Fa Zhang, Yi Ru-Ya Zhou, Yuan Analysis of PDEs 35J61 Let $ 0\le f\in C^{0,1}(\mathbb R^n)$. Given a domain $Ω\subset \mathbb R^n$, we prove that any stable solution to the equation $-Δu=f(u)$ in $Ω$ satisfies a BMO interior regularity when $n=10$, and an Morrey $M^{p_n,4+2/(p_n-2)}$ interior regularity when $n\ge 11$, where $$p_n=\frac{2(n-2\sqrt{n-1}-2)}{n-2\sqrt{n-1}-4}. $$ This result is optimal as hinted by earlier results, and answers an open question raised by Cabré, Figalli, Ros-Oton and Serra. As an application, we show a sharp Liouville property: Any stable solution $u \in C^2(\mathbb R^n)$ to $-Δu=f(u)$ in $\mathbb R^n$ satisfying the growth condition, i.e.\ $|u(x)|= o\left( \log|x| \right)$ as $|x|\to+\infty$ when $n=10$; or $|u(x)|= o\left( |x| ^{ -\frac n2+\sqrt{n-1}+2 }\right)$ as $x|\to+\infty$ when $n\ge 11$, must be a constant. This extends the well-known Liouville property for radial stable solutions obtained by Villegas. |
| title | Optimal regularity & Liouville property for stable solutions to semilinear elliptic equations in $\mathbb R^n$ with $n\ge10$ |
| topic | Analysis of PDEs 35J61 |
| url | https://arxiv.org/abs/2105.02535 |