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

Abstract

Given a smoothly bounded domain $\Omega ⋐{\mathbb{R}}^n$ with $n\ge 1$ odd, we study the blow-up of bounded sequences $(u_k)\subset H_{00}^{\frac{n}{2}}(\Omega )$ of solutions to the non-local equation $$(-\Delta {)}^{\frac{n}{2}}{u}_k={\lambda }_k{u}_k{e}^{\frac{n}{2}{u}_k^2}\text{in }\Omega ,$$ where ${\lambda }_k\to {\lambda }_{\infty }\in [0,\infty )$, and $H_{00}^{\frac{n}{2}}(\Omega )$ 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 $$(-\Delta {)}^{\frac{n}{2}}\eta =(n-1)!e^{n\eta }\text{in }{\mathbb{R}}^n$$ 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 $\Lambda :={\mathrm{limsup}}_{k\to \infty }\Vert (-\Delta {)}^{\frac{n}{4}}{u}_k{\Vert }_{L^2}^2\ge {\Lambda }_1:=(n-1)!|S^n|$.

References in corpus (3)

• Fractional Adams-Moser-Trudinger type inequalities
• Blow-up analysis of a nonlocal Liouville-type equation
• n/p-Harmonic maps: regularity for the sphere case

Cited by in corpus (5)

• Fractional Adams-Moser-Trudinger type inequalities
• Improved Adams-type inequalities and their extremals in dimension 2m
• Regularity for a fractional p-Laplace equation
• Boundary regularity for conformally invariant variational problems with Neumann data
• A sharp trace Adams' inequality in ${\mathbb{R}}^4$ and Existence of the extremals