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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2507.09533 |
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| _version_ | 1866913939179175936 |
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| author | Carl, Siegfried Perera, Kanishka Tehrani, Hossein |
| author_facet | Carl, Siegfried Perera, Kanishka Tehrani, Hossein |
| contents | In this paper we present a new global $L^\infty$-estimate for solutions $u\in D^{s,p}(\R^N)$ of the fractional $p$-Laplacian equation %
$$ u\in D^{s,p}(\R^N): (-Δ_p)^s u=f(x,u) \quad\mbox{in }\R^N, $$ %
of the form %
$$ \|u\|_{\infty}\le C Φ(\|u\|_β) $$ %
for some $β> p$, where $Φ: \R^+\to \R^+$ is a data independent function with $\lim_{s\to 0^+}Φ(s)=0$. The obtained $L^\infty$-estimate is used to prove a decay estimate based on pointwise estimates in terms of nonlinear Wolff potentials. Taking advantage of both the $L^\infty$ and decay estimate we prove a Brezis-Nirenberg type result regarding $D^{s,2}(\R^N)$ versus $C_b\left(\R^N, 1+|x|^{N-2s}\right)$ local minimizers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09533 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global $L^\infty$ and decay estimate for fractional $p$-Laplacian equations in $D^{s,p}(\R^N)$ Carl, Siegfried Perera, Kanishka Tehrani, Hossein Analysis of PDEs 35B38, 35B40, 35B45, 35B51, 35J20, 35J60, 31C05 In this paper we present a new global $L^\infty$-estimate for solutions $u\in D^{s,p}(\R^N)$ of the fractional $p$-Laplacian equation % $$ u\in D^{s,p}(\R^N): (-Δ_p)^s u=f(x,u) \quad\mbox{in }\R^N, $$ % of the form % $$ \|u\|_{\infty}\le C Φ(\|u\|_β) $$ % for some $β> p$, where $Φ: \R^+\to \R^+$ is a data independent function with $\lim_{s\to 0^+}Φ(s)=0$. The obtained $L^\infty$-estimate is used to prove a decay estimate based on pointwise estimates in terms of nonlinear Wolff potentials. Taking advantage of both the $L^\infty$ and decay estimate we prove a Brezis-Nirenberg type result regarding $D^{s,2}(\R^N)$ versus $C_b\left(\R^N, 1+|x|^{N-2s}\right)$ local minimizers. |
| title | Global $L^\infty$ and decay estimate for fractional $p$-Laplacian equations in $D^{s,p}(\R^N)$ |
| topic | Analysis of PDEs 35B38, 35B40, 35B45, 35B51, 35J20, 35J60, 31C05 |
| url | https://arxiv.org/abs/2507.09533 |