Discrete Caffarelli-Kohn-Nirenberg inequalities and ground state solutions to nonlinear elliptic equations

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]: $$\parallel u{\parallel }_{{\mathrm{\ell }}_b^q}\le C(a,b,c,p,q,r,\theta ,N)\parallel u{\parallel }_{D_a^{1,p}}^{\theta }\parallel u{\parallel }_{{\mathrm{\ell }}_c^r}^{1-\theta },\mathrm{\forall }u\in D_{a,0}^{1,p}({\mathbb{Z}}^N)\cap {\mathrm{\ell }}_c^r({\mathbb{Z}}^N),$$ 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^{\ast }}+\frac{b}{N}=\theta (\frac{1}{p}+\frac{a-1}{N})+(1-\theta )(\frac{1}{r}+\frac{c}{N})$ and $q\ge q^{\ast }$. 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 $q>q^{\ast }$. As an application, we get positive ground state solutions to the nonlinear elliptic equations.