Asymptotic expansions for conformal scalar curvature equations near isolated singularities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910343963344896 |
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| author | Du, Xusheng Yang, Hui |
| author_facet | Du, Xusheng Yang, Hui |
| contents | In this paper, we study asymptotic expansions of positive solutions of the conformal scalar curvature equation $$ - Δu = K(x) u^\frac{n + 2}{n - 2} ~~~~~~ \textmd{in} ~ B_1 \setminus \{ 0 \} $$ with an isolated singularity at the origin. Under certain flatness conditions on $K$, we establish a higher-order expansion of solutions near the origin. In particular, we give the refined second-order asymptotic expansion of solutions when $n \geq 6$. Moreover, we also obtain an arbitrary-order expansion of singular positive solutions of the anisotropic elliptic equation $$ - \,{\rm div} (|x|^{- 2 a} \nabla u) = |x|^{- b p} u^{p - 1} ~~~~~~ \textmd{in} ~ B_1 \setminus \{ 0 \}, $$ where $0 \leq a < \frac{n - 2}{2}$, $a \leq b < a + 1$ and $p = \frac{2 n}{n - 2 + 2 (b - a)}$. This equation is arising from the celebrated Caffarelli-Kohn-Nirenberg inequality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_16597 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic expansions for conformal scalar curvature equations near isolated singularities Du, Xusheng Yang, Hui Analysis of PDEs 35J61, 35C20 In this paper, we study asymptotic expansions of positive solutions of the conformal scalar curvature equation $$ - Δu = K(x) u^\frac{n + 2}{n - 2} ~~~~~~ \textmd{in} ~ B_1 \setminus \{ 0 \} $$ with an isolated singularity at the origin. Under certain flatness conditions on $K$, we establish a higher-order expansion of solutions near the origin. In particular, we give the refined second-order asymptotic expansion of solutions when $n \geq 6$. Moreover, we also obtain an arbitrary-order expansion of singular positive solutions of the anisotropic elliptic equation $$ - \,{\rm div} (|x|^{- 2 a} \nabla u) = |x|^{- b p} u^{p - 1} ~~~~~~ \textmd{in} ~ B_1 \setminus \{ 0 \}, $$ where $0 \leq a < \frac{n - 2}{2}$, $a \leq b < a + 1$ and $p = \frac{2 n}{n - 2 + 2 (b - a)}$. This equation is arising from the celebrated Caffarelli-Kohn-Nirenberg inequality. |
| title | Asymptotic expansions for conformal scalar curvature equations near isolated singularities |
| topic | Analysis of PDEs 35J61, 35C20 |
| url | https://arxiv.org/abs/2402.16597 |