Maalaoui, Ali and Martinazzi, Luca and Schikorra, Armin. (2016) Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension. Communications Partial Differential Equations, 41 (10). pp. 1593-1618.
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Official URL: http://edoc.unibas.ch/45255/
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Abstract
Given a smoothly bounded domain \Omega\Subset\mathbb{R}^n with n\ge 1 odd, we study the blow-up of bounded sequences (u_k)\subset H^\frac{n}{2}_{00}(\Omega) of solutions to the non-local equation
(-\Delta)^\frac n2 u_k=\lambda_k u_ke^{\frac n2 u_k^2}\quad \text{in
}\Omega,
where \lambda_k\to\lambda_\infty \in [0,\infty), and H^{\frac n2}_{00}(\Omega) denotes the Lions-Magenes spaces of functions u\in L^2(\mathbb{R}^n) which are supported in \Omega and with (-\Delta)^\frac{n}{4}u\in L^2(\mathbb{R}^n). Extending previous works of Druet, Robert-Struwe and the second author, we show that if the sequence (u_k) is not bounded in L^\infty(\Omega), a suitably rescaled subsequence \eta_k converges to the function \eta_0(x)=\log\left(\frac{2}{1+|x|^2}\right), which solves the prescribed non-local Q-curvature equation
(-\Delta)^\frac n2 \eta
=(n-1)!e^{n\eta}\quad \text{in }\mathbb{R}^n
recently studied by Da Lio-Martinazzi-Rivi\`ere when n=1, Jin-Maalaoui-Martinazzi-Xiong when n=3, and Hyder when n\ge 5 is odd. We infer that blow-up can occur only if \Lambda:=\limsup_{k\to \infty}\|(-\Delta)^\frac n4 u_k\|_{L^2}^2\ge \Lambda_1:= (n-1)!|S^n|.
(-\Delta)^\frac n2 u_k=\lambda_k u_ke^{\frac n2 u_k^2}\quad \text{in
}\Omega,
where \lambda_k\to\lambda_\infty \in [0,\infty), and H^{\frac n2}_{00}(\Omega) denotes the Lions-Magenes spaces of functions u\in L^2(\mathbb{R}^n) which are supported in \Omega and with (-\Delta)^\frac{n}{4}u\in L^2(\mathbb{R}^n). Extending previous works of Druet, Robert-Struwe and the second author, we show that if the sequence (u_k) is not bounded in L^\infty(\Omega), a suitably rescaled subsequence \eta_k converges to the function \eta_0(x)=\log\left(\frac{2}{1+|x|^2}\right), which solves the prescribed non-local Q-curvature equation
(-\Delta)^\frac n2 \eta
=(n-1)!e^{n\eta}\quad \text{in }\mathbb{R}^n
recently studied by Da Lio-Martinazzi-Rivi\`ere when n=1, Jin-Maalaoui-Martinazzi-Xiong when n=3, and Hyder when n\ge 5 is odd. We infer that blow-up can occur only if \Lambda:=\limsup_{k\to \infty}\|(-\Delta)^\frac n4 u_k\|_{L^2}^2\ge \Lambda_1:= (n-1)!|S^n|.
Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Mathematik 05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Analysis (Martinazzi) |
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UniBasel Contributors: | Martinazzi, Luca |
Item Type: | Article, refereed |
Article Subtype: | Research Article |
Publisher: | Taylor & Francis |
ISSN: | 0360-5302 |
e-ISSN: | 1532-4133 |
Note: | Publication type according to Uni Basel Research Database: Journal article |
Identification Number: | |
Last Modified: | 02 Nov 2017 07:44 |
Deposited On: | 02 Nov 2017 07:44 |
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