Calculus of Variations and Geometric Measure Theory
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A. Maalaoui - L. Martinazzi - A. Schikorra

Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension

created by martinazz on 17 Jul 2018



Inserted: 17 jul 2018

Year: 2015

ArXiv: 1504.00254 PDF


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 $$(-\Delta)\frac n2 uk=\lambdak uke{\frac n2 uk2}\quad \text{in }\Omega,$$ 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 $\eta_0(x)=\log\left(\frac{2}{1+
^2}\right)$, 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)!

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