Optimal decay of extremals for the fractional Sobolev inequality (Q283534)

For \(s\in (0,1)\), \(p>1\), \(N>sp\), define \[ D^{s,p}(\mathbb R^N)=\left\{u\in L^{\frac{Np}{N-sp}}(\mathbb R^N)| \int_{\mathbb R^{2n}} \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx \,dy<\infty \right\}, \] and consider \[ S_{p,s}=\underset{u\in D^{s,p}(\mathbb R^N) \setminus \{0\}} \inf \frac{\int_{\mathbb R^{2n}} \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx \,dy}{\int_{\mathbb R^N }|u|^{\frac{Np}{N-sp}} \,dx}.\eqno(1) \] It is proved that, if \(U\in D^{s,p}(\mathbb R^N)\) is any minimizer for (1), then \(U\in L^{\infty}(\mathbb R^N)\) is a constant sign, radially symmetric and monotone function with \[ \lim_{|x|\to\infty} |x|^{\frac{N-sp}{p-1}} U(x)=U_{\infty} \] for some constant \(U_{\infty}\in \mathbb R\setminus \{0\}\) (Theorem 1.1). The authors point out the relation between the above result and the proof of the existence of weak solutions for the non-local Brezis-Nirenberg problem in a smooth bounded domain \(\Omega\subset \mathbb R^N,\) i.e., \[ \begin{aligned} (-\Delta_p)^su&=\lambda |u|^{p-2}u+|u|^{\frac{Np}{N-sp}-2}u\;\text{ in }\Omega, \\ u&=0\;\text{ in } \mathbb R^N\setminus \Omega,\end{aligned} \] where \(\lambda\) is positive, and \(\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\). A rigorous computation of the fractional \(p\)-Laplace operator of a power function is presented in Appendix A of the paper.