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

Abstract

In this paper, we prove the discrete Caffarelli-Kohn-Nirenberg inequalities on the lattice $\mathbb{Z}^{N}$ ($N\geq 1$) in a broader range of parameters than the classical continuous version [8]: \[ \parallel u\parallel_{\ell_{b}^{q}}\leq C(a,b,c,p,q,r,\theta,N)\parallel u\parallel_{D_{a}^{1,p}}^{\theta}\parallel u\parallel_{\ell_{c}^{r}}^{1-\theta},\:\forall u\in D_{a,0}^{1,p}(\mathbb{Z}^{N}) \cap \ell_c ^r(\mathbb{Z}^{N}), \] where $p,q,r>1,0\leq\theta\leq1$, $\frac{1}{p}+\frac{a}{N}>0,\frac{1}{r}+\frac{c}{N}>0,b\leq\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 $q\geq q^{\ast}$. 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.