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

TL;DR

This paper establishes discrete versions of the Caffarelli-Kohn-Nirenberg inequalities on integer lattices, broadening the parameter range and proving the existence of extremal functions and ground state solutions for related nonlinear elliptic equations.

Contribution

It extends classical inequalities to discrete settings on lattices and proves the existence of extremal functions and ground state solutions in this broader context.

Findings

Discrete Caffarelli-Kohn-Nirenberg inequalities are valid on $ extbf{Z}^N$ for a wider parameter range.

Existence of extremal functions for the best constants in supercritical cases is established.

Positive ground state solutions to nonlinear elliptic equations are obtained as an application.

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.