In this paper we deal with the topic in two parts. First, we are interested in discussing whether <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ \theta _{1} \in \mathbb {S}_{\mathcal {H}}^{\varpi ; \psi }\left( \Omega \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>θ</mi> <mn>1</mn> </msub> <mo>∈</mo> <msubsup> <mi mathvariant="double-struck">S</mi> <mrow> <mi mathvariant="script">H</mi> </mrow> <mrow> <mi>ϖ</mi> <mo>;</mo> <mi>ψ</mi> </mrow> </msubsup> <mfenced close=")" open="("> <mi mathvariant="normal">Ω</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> solves the fractional p-Laplacian equation, then either <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\theta _{1}>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>θ</mi> <mn>1</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\theta _{1}<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>θ</mi> <mn>1</mn> </msub> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. It is also extremely important to ensure that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f^{p} \phi \in \mathbb {S}_{\mathcal {H}}^{\varpi ; \psi }\left( \Omega \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>f</mi> <mi>p</mi> </msup> <mi>ϕ</mi> <mo>∈</mo> <msubsup> <mi mathvariant="double-struck">S</mi> <mrow> <mi mathvariant="script">H</mi> </mrow> <mrow> <mi>ϖ</mi> <mo>;</mo> <mi>ψ</mi> </mrow> </msubsup> <mfenced close=")" open="("> <mi mathvariant="normal">Ω</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. In this sense, motivated by the previous results, we are interested in proving the radial symmetry eigenvalue of the p-Laplacian fractional equation in space <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation>-fractional <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ \mathbb {S}_{\mathcal {H}}^{\varpi ; \psi }\left( \Omega \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="double-struck">S</mi> <mrow> <mi mathvariant="script">H</mi> </mrow> <mrow> <mi>ϖ</mi> <mo>;</mo> <mi>ψ</mi> </mrow> </msubsup> <mfenced close=")" open="("> <mi mathvariant="normal">Ω</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$1 <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\varpi \in \left( 0,1 \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϖ</mi> <mo>∈</mo> <mfenced close=")" open="("> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$p \varpi <n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mi>ϖ</mi> <mo><</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. Finally, we present two examples and comments on possible applications of the problem (<InternalRef RefID="Equ1">1.1</InternalRef>).

Radial symmetry and fractional p-Laplacian equation

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2026年2月3日

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J. Vanterler da C. Sousa,
J. C. A. Soares,
F. S. Costa

摘要

In this paper we deal with the topic in two parts. First, we are interested in discussing whether $ \theta _{1} \in \mathbb {S}_{\mathcal {H}}^{\varpi ; \psi }\left( \Omega \right) $ $θ_{1} \in S_{H}^{ϖ; ψ} (Ω)$ solves the fractional p-Laplacian equation, then either $\theta _{1}>0$ $θ_{1} > 0$ or $\theta _{1}<0$ $θ_{1} < 0$ in $\Omega $ $Ω$ . It is also extremely important to ensure that $f^{p} \phi \in \mathbb {S}_{\mathcal {H}}^{\varpi ; \psi }\left( \Omega \right) $ $f^{p} ϕ \in S_{H}^{ϖ; ψ} (Ω)$ . In this sense, motivated by the previous results, we are interested in proving the radial symmetry eigenvalue of the p-Laplacian fractional equation in space $\psi $ $ψ$ -fractional $ \mathbb {S}_{\mathcal {H}}^{\varpi ; \psi }\left( \Omega \right) $ $S_{H}^{ϖ; ψ} (Ω)$ with $1< p < \infty $ $1 < p < \infty$ , $\varpi \in \left( 0,1 \right) $ $ϖ \in (0, 1)$ and $p \varpi <n$ $p ϖ < n$ . Finally, we present two examples and comments on possible applications of the problem (1.1).