2 results
A sharp threshold for Trudinger–Moser type inequalities with logarithmic kernels in dimension N
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 17 February 2025, pp. 1-39
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In the article, we investigate Trudinger–Moser type inequalities in presence of logarithmic kernels in dimension N. A sharp threshold, depending on N, is detected for the existence of extremal functions or blow-up, where the domain is the ball or the entire space
$\mathbb{R}^N$. We also show that the extremal functions satisfy suitable Euler–Lagrange equations. When the domain is the entire space, such equations can be derived by a N-Laplacian Schrödinger equation strongly coupled with a higher order fractional Poisson’s equation. The results extends [16] to any dimension 
$N \geq 2$.
General sharp weighted Caffarelli–Kohn–Nirenberg inequalities
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 149 / Issue 3 / June 2019
- Published online by Cambridge University Press:
- 27 December 2018, pp. 691-718
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- June 2019
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In this paper, we will use optimal mass transport combining with suitable transforms to study the sharp constants and optimizers for a class of the Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities. Moreover, we will investigate these inequalities with and without the monomial weights
$x_{1}^{A_{1}} \cdots x_{N}^{A_{N}}$ on ℝN.
