Moving sphere approach to a general weighted integral equation
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910840736710656 |
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| author | Lê, Quynh N. T. Nguyen, Tien-Tai |
| author_facet | Lê, Quynh N. T. Nguyen, Tien-Tai |
| contents | Let $p$ be positive and $n \geq 3$ be an integer. Let $f(\cdot,\cdot): \mathbf{R}_+\times \mathbf{R}_+\to \mathbf{R}_+$ be a continuous function. In this paper, we are concerned with positive solutions to the following integral equation \[ u(x)= \int_{\mathbf{R}^n} |x-y|^p f(|y|,u(y)) dy \quad\text{in }\mathbf{R}^n\setminus\{\textbf{0}\}. \] By imposing some suitable conditions on $f$, we obtain the radially symmetry property of positive solutions to the above equation by using the method of moving spheres in integral form. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_16039 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moving sphere approach to a general weighted integral equation Lê, Quynh N. T. Nguyen, Tien-Tai Analysis of PDEs 26D15, 35B06, 35C15, 35J30, 58J70 Let $p$ be positive and $n \geq 3$ be an integer. Let $f(\cdot,\cdot): \mathbf{R}_+\times \mathbf{R}_+\to \mathbf{R}_+$ be a continuous function. In this paper, we are concerned with positive solutions to the following integral equation \[ u(x)= \int_{\mathbf{R}^n} |x-y|^p f(|y|,u(y)) dy \quad\text{in }\mathbf{R}^n\setminus\{\textbf{0}\}. \] By imposing some suitable conditions on $f$, we obtain the radially symmetry property of positive solutions to the above equation by using the method of moving spheres in integral form. |
| title | Moving sphere approach to a general weighted integral equation |
| topic | Analysis of PDEs 26D15, 35B06, 35C15, 35J30, 58J70 |
| url | https://arxiv.org/abs/2502.16039 |