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

Abstract

Given a smoothly bounded domain $Ω\Subset\mathbb{R}^n$ with $n\ge 1$ odd, we study the blow-up of bounded sequences $(u_k)\subset H^\frac{n}{2}_{00}(Ω)$ of solutions to the non-local equation $$(-Δ)^\frac n2 u_k=λ_k u_ke^{\frac n2 u_k^2}\quad \text{in }Ω,$$ where $λ_k\toλ_\infty \in [0,\infty)$, and $H^{\frac n2}_{00}(Ω)$ denotes the Lions-Magenes spaces of functions $u\in L^2(\mathbb{R}^n)$ which are supported in $Ω$ and with $(-Δ)^\frac{n}{4}u\in L^2(\mathbb{R}^n)$. Extending previous works of Druet, Robert-Struwe and the second author, we show that if the sequence $(u_k)$ is not bounded in $L^\infty(Ω)$, a suitably rescaled subsequence $η_k$ converges to the function $η_0(x)=\log\left(\frac{2}{1+|x|^2}\right)$, which solves the prescribed non-local $Q$-curvature equation $$(-Δ)^\frac n2 η=(n-1)!e^{nη}\quad \text{in }\mathbb{R}^n$$ recently studied by Da Lio-Martinazzi-Rivière 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 $Λ:=\limsup_{k\to \infty}\|(-Δ)^\frac n4 u_k\|_{L^2}^2\ge Λ_1:= (n-1)!|S^n|$.