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

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|.

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