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Optimal decay of extremals for the fractional Sobolev inequality. (English) Zbl 1350.46024
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.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35B40 Asymptotic behavior of solutions to PDEs 49K20 Optimality conditions for problems involving partial differential equations 35J92 Quasilinear elliptic equations with $$p$$-Laplacian
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