A Detailed Analysis on Sharpened Singular Adams-Type Inequalities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915719407468544 |
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| author | Mahanta, Deepak Kumar Mukherjee, Tuhina Sarkar, Abhishek |
| author_facet | Mahanta, Deepak Kumar Mukherjee, Tuhina Sarkar, Abhishek |
| contents | We establish a sharp Adams-type inequality in higher-order function spaces with singular weights on $\mathbb{R}^n$. A sharp singular concentration-compactness principle, improving Lions' result, is also proved. The study distinguishes between critical and subcritical sharp singular Adams-type inequalities and shows their equivalence. Furthermore, we analyze the asymptotic behavior of the associated bounds and relate the suprema of the critical and subcritical cases. A new compact embedding, crucial to our analysis, is also derived. Moreover, as an application of these results, by employing the mountain pass theorem, we study the existence of nontrivial solutions to a class of nonhomogeneous quasilinear elliptic equations involving the $(p,\frac{n}{2})$-biharmonic operator with singular exponential growth. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_11176 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Detailed Analysis on Sharpened Singular Adams-Type Inequalities Mahanta, Deepak Kumar Mukherjee, Tuhina Sarkar, Abhishek Analysis of PDEs 35A23, 35B33, 35J30, 26D10 We establish a sharp Adams-type inequality in higher-order function spaces with singular weights on $\mathbb{R}^n$. A sharp singular concentration-compactness principle, improving Lions' result, is also proved. The study distinguishes between critical and subcritical sharp singular Adams-type inequalities and shows their equivalence. Furthermore, we analyze the asymptotic behavior of the associated bounds and relate the suprema of the critical and subcritical cases. A new compact embedding, crucial to our analysis, is also derived. Moreover, as an application of these results, by employing the mountain pass theorem, we study the existence of nontrivial solutions to a class of nonhomogeneous quasilinear elliptic equations involving the $(p,\frac{n}{2})$-biharmonic operator with singular exponential growth. |
| title | A Detailed Analysis on Sharpened Singular Adams-Type Inequalities |
| topic | Analysis of PDEs 35A23, 35B33, 35J30, 26D10 |
| url | https://arxiv.org/abs/2412.11176 |