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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| Accesso online: | https://arxiv.org/abs/2211.10554 |
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| _version_ | 1866911078652313600 |
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| author | Chen, Huyuan Peng, Huihuan Sun, Yanqing |
| author_facet | Chen, Huyuan Peng, Huihuan Sun, Yanqing |
| contents | Our purpose of this paper is to investigate positive solutions of the elliptic equation with regional fractional Laplacian $$
( - Δ)_{B_1}^s u +u= h(x,u) \quad
{\rm in} \ \, B_1,\qquad u\in C_0(B_1),
$$
where $( - Δ)_{B_1}^s$ with $s\in(0,\frac12]$ is the regional fractional Laplacian and $h$ is the nonlinearity.
Ordinarily, positive solutions vanishing at the boundary are not anticipated to be derived for the equations with regional fractional Laplacian of order $s\in(0,\frac12]$. Positive solutions are obtained when the nonlinearity assumes the following two models:
$h(x,t)=f(x)$ or $h(x,t)=h_1(x)\, t^p+ εh_2(x)$, where $p>1$, $ε>0$ small and $f, h_1, h_2$ are Hölder continuous, radially symmetric and decreasing functions under suitable conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_10554 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Existence of solutions to elliptic equations involving regional fractional Laplacian with order $(0,\frac12]$ Chen, Huyuan Peng, Huihuan Sun, Yanqing Analysis of PDEs Our purpose of this paper is to investigate positive solutions of the elliptic equation with regional fractional Laplacian $$ ( - Δ)_{B_1}^s u +u= h(x,u) \quad {\rm in} \ \, B_1,\qquad u\in C_0(B_1), $$ where $( - Δ)_{B_1}^s$ with $s\in(0,\frac12]$ is the regional fractional Laplacian and $h$ is the nonlinearity. Ordinarily, positive solutions vanishing at the boundary are not anticipated to be derived for the equations with regional fractional Laplacian of order $s\in(0,\frac12]$. Positive solutions are obtained when the nonlinearity assumes the following two models: $h(x,t)=f(x)$ or $h(x,t)=h_1(x)\, t^p+ εh_2(x)$, where $p>1$, $ε>0$ small and $f, h_1, h_2$ are Hölder continuous, radially symmetric and decreasing functions under suitable conditions. |
| title | Existence of solutions to elliptic equations involving regional fractional Laplacian with order $(0,\frac12]$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2211.10554 |