Stable solutions to fractional semilinear equations: uniqueness, classification, and approximation results
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917927852179456 |
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| author | Sanz-Perela, Tomás |
| author_facet | Sanz-Perela, Tomás |
| contents | We study stable solutions to fractional semilinear equations $(-Δ)^s u = f(u)$ in $Ω\subset \mathbb{R}^n$, for convex nonlinearities $f$, and under the Dirichlet exterior condition $u=g$ in $\mathbb{R}^n \setminus Ω$ with general $g$. We establish a uniqueness and a classification result, and we show that weak (energy) stable solutions can be approximated by a sequence of bounded (and hence regular) stable solutions to similar problems.
As an application of our results, we establish the interior regularity of weak (energy) stable solutions to the problem for the half-Laplacian in dimensions $1 \leq n \leq 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_02477 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Stable solutions to fractional semilinear equations: uniqueness, classification, and approximation results Sanz-Perela, Tomás Analysis of PDEs 35J61, 35R11, 35B35, 35B65 We study stable solutions to fractional semilinear equations $(-Δ)^s u = f(u)$ in $Ω\subset \mathbb{R}^n$, for convex nonlinearities $f$, and under the Dirichlet exterior condition $u=g$ in $\mathbb{R}^n \setminus Ω$ with general $g$. We establish a uniqueness and a classification result, and we show that weak (energy) stable solutions can be approximated by a sequence of bounded (and hence regular) stable solutions to similar problems. As an application of our results, we establish the interior regularity of weak (energy) stable solutions to the problem for the half-Laplacian in dimensions $1 \leq n \leq 4$. |
| title | Stable solutions to fractional semilinear equations: uniqueness, classification, and approximation results |
| topic | Analysis of PDEs 35J61, 35R11, 35B35, 35B65 |
| url | https://arxiv.org/abs/2210.02477 |