Quantitative Stability in Fractional Hardy-Sobolev Inequalities: The Role of Euler-Lagrange Equations
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866908720548544512 |
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| author | Chakraborty, Souptik Sarkar, Utsab |
| author_facet | Chakraborty, Souptik Sarkar, Utsab |
| contents | This paper investigates sharp stability estimates for the fractional Hardy-Sobolev inequality: $$μ_{s,t}\left(\mathbb{R}^N\right) \left(\int_{\mathbb{R}^N} \frac{|u|^{2^*_s(t)}}{|x|^t} \,{\rm d}x \right)^{\frac{2}{2^*_s(t)}} \leq \int_{\mathbb{R}^N} \left|(-Δ)^{\frac{s}{2}} u \right|^2 \,{\rm d}x, \quad \text{for all } u \in \dot{H}^s\left(\mathbb{R}^N\right),$$ where $N > 2s$, $s \in (0,1)$, $0 < t < 2s < N $, and $2^*_s(t) = \frac{2(N-t)}{N-2s}$. Here, $μ_{s,t}\left(\mathbb{R}^N\right)$ represents the best constant in the inequality.
The paper focuses on the quantitative stability results of the above inequality and the corresponding Euler-Lagrange equation near a positive ground-state solution. Additionally, a qualitative stability result is established for the Euler-Lagrange equation, offering a thorough characterization of the Palais-Smale sequences for the associated energy functional. These results generalize the sharp quantitative stability results for the classical Sobolev inequality in $\mathbb{R}^N$, originally obtained by Bianchi and Egnell \cite{BE91} as well as the corresponding critical exponent problem in $\mathbb{R}^N$, explored by Ciraolo, Figalli, and Maggi \cite{CFM18} in the framework of fractional calculus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06716 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative Stability in Fractional Hardy-Sobolev Inequalities: The Role of Euler-Lagrange Equations Chakraborty, Souptik Sarkar, Utsab Analysis of PDEs 35A23, 35R11, 35B33, 35B35 This paper investigates sharp stability estimates for the fractional Hardy-Sobolev inequality: $$μ_{s,t}\left(\mathbb{R}^N\right) \left(\int_{\mathbb{R}^N} \frac{|u|^{2^*_s(t)}}{|x|^t} \,{\rm d}x \right)^{\frac{2}{2^*_s(t)}} \leq \int_{\mathbb{R}^N} \left|(-Δ)^{\frac{s}{2}} u \right|^2 \,{\rm d}x, \quad \text{for all } u \in \dot{H}^s\left(\mathbb{R}^N\right),$$ where $N > 2s$, $s \in (0,1)$, $0 < t < 2s < N $, and $2^*_s(t) = \frac{2(N-t)}{N-2s}$. Here, $μ_{s,t}\left(\mathbb{R}^N\right)$ represents the best constant in the inequality. The paper focuses on the quantitative stability results of the above inequality and the corresponding Euler-Lagrange equation near a positive ground-state solution. Additionally, a qualitative stability result is established for the Euler-Lagrange equation, offering a thorough characterization of the Palais-Smale sequences for the associated energy functional. These results generalize the sharp quantitative stability results for the classical Sobolev inequality in $\mathbb{R}^N$, originally obtained by Bianchi and Egnell \cite{BE91} as well as the corresponding critical exponent problem in $\mathbb{R}^N$, explored by Ciraolo, Figalli, and Maggi \cite{CFM18} in the framework of fractional calculus. |
| title | Quantitative Stability in Fractional Hardy-Sobolev Inequalities: The Role of Euler-Lagrange Equations |
| topic | Analysis of PDEs 35A23, 35R11, 35B33, 35B35 |
| url | https://arxiv.org/abs/2503.06716 |