In this paper, we prove the discrete Caffarelli-Kohn-Nirenberg inequalities on the lattice \(\mathbb {Z}^{N}\) ( \(N\ge 1\) ) in a broader range of parameters than the classical continuous version [8]: \( \Vert u\Vert _{\ell _{b}^{q}}\le C(a,b,c,p,q,r,\theta ,N)\Vert u\Vert _{D_{a}^{1,p}}^{\theta }\Vert u\Vert _{\ell _{c}^{r}}^{1-\theta },\,\forall u\in D_{a,0}^{1,p}(\mathbb {Z}^{N}) \cap \ell _c ^r(\mathbb {Z}^{N}), \) where \(p,q,r>1,0\le \theta \le 1\) , \(\frac{1}{p}+\frac{a}{N}>0,\frac{1}{r}+\frac{c}{N}>0,b\le \theta a+(1-\theta )c,\) \(\frac{1}{q^{*}}+\frac{b}{N}= \theta (\frac{1}{p}+\frac{a-1}{N})+(1-\theta )(\frac{1}{r}+\frac{c}{N})\) and \(q\ge q^{*}\) . For two special cases \(\theta =1,a=0\) and \(a=b=c=0\) , by the discrete Schwarz rearrangement established in [24], we prove the existence of extremal functions for the best constants in the supercritical case \(q>q^{*}\) . As an application, we get positive ground state solutions to the nonlinear elliptic equations.
