Optimizers of the Sobolev and Gagliardo–Nirenberg inequalities in $\dot{W}^{s,p}$

Abstract

In this paper, we consider the existence of optimizers for the following Sobolev and Gagliardo–Nirenberg intepolation inequalities in $\dot{W}^{s,p}({\mathbb {R}}^d)$ at the non-endpoint case:

$$\begin{aligned} ||u||_{L^{p^*}} \le C || u||_{ \dot{W}^{s,p}}, \qquad ||u||_{L^r} \le C || u||_{\dot{W}^{s_1,p}}^{\theta } ||u||_{L^p}^{1-\theta }, \end{aligned}$$

where $\dot{W}^{s,p} ({\mathbb {R}}^d)$ is the fractional Sobolev space, defined by

$$\begin{aligned} \dot{W}^{s,p}({\mathbb {R}}^d) =\bigg \{ u\in L^1_{loc}: \frac{u(x)-u(y)}{|x-y|^{s+\frac{d}{p}}} \in L^p({\mathbb {R}}^d \times {\mathbb {R}}^d) \bigg \}, \end{aligned}$$

and $0<s,s_1<1,$ $1<p,r <\infty ,$ $\frac{1}{p^*}=\frac{1}{p}-\frac{s}{d}>0,$ $\frac{1}{r}=\theta (\frac{1}{p}-\frac{s_1}{d}) +(1-\theta )\frac{1}{p} .$ Comparing with the usual assumption, we don’t need $s_1 < \frac{d}{p},$ i.e., $p<r < \frac{pd}{d-s_1p}$ if $\frac{1}{p}-\frac{s_1}{d} >0,$ $p<r <\infty ,$ if $\frac{1}{p}-\frac{s_1}{d} \le 0.$ To prove these, we establish a compactness up to symmetry lemma in Sobolev embedding for $\dot{W}^{s,p}$ case by refined Sobolev inequalities in Morrey or Besov space, and we handle the two cases in a unified way. Also for existence of optimizers for Gagliardo–Nirenberg intepolation inequalities in $\dot{W}^{s,p} \cap L^p,$ we give two alternative proofs: one follows the celebrated paper (Bellazzini et al. in Math Ann 360(3–4):653–673, 2014. https://doi.org/10.1007/s00208-014-1046-2), in which we establish a compactness up to translation lemma in $\dot{W}^{s,p} \cap L^p;$ one follows the concentrarion compactness principle (Lions in Ann Inst Henri Poincaré Anal Non Linéaire 1(2):109–145, 1984), (Lions in Ann Inst Henri Poincaré Anal Non Linéaire 1(4):223–283 1984).