Sharp Born--Infeld Sobolev inequality and extremal functions on the lattice graph ${\mathbb{Z}}^N$

In this paper, we study the sharp Sobolev-type inequality associated with the Born–Infeld energy on the lattice graph ${\mathbb{Z}}^N$, $N\ge 3$: $$\frac{1}{2}\sum _{x\in {\mathbb{Z}}^N}\sum _{y\sim x}(1-\sqrt{1-\mathrm{\mid }{\mathrm{\nabla }}_{xy}u{\mathrm{\mid }}^2})\ge C_{N,\alpha }{(\sum _{x\in {\mathbb{Z}}^N}\mathrm{\mid }u(x){\mathrm{\mid }}^{\alpha })}^{\frac{N}{N+\alpha }},$$ for $u\in D^{1,2}({\mathbb{Z}}^N)\cap {\mathrm{\ell }}^{\alpha }({\mathbb{Z}}^N)$ satisfying $\mathrm{\mid }{\mathrm{\nabla }}_{xy}u\mathrm{\mid }\le 1$ for every $x\sim y$, where $C_{N,\alpha }$ denotes the optimal constant. We determine the exact positivity threshold and prove that $C_{N,\alpha }>0$ if and only if $\alpha \ge 2^{\ast }:=2N\mathrm{/}(N-2)$. At the Sobolev critical exponent $\alpha =2^{\ast }$, we identify the optimal constant as $$C_{N,2^{\ast }}=\frac{1}{2}{\mathcal{S}}_2,$$ where ${\mathcal{S}}_2$ is the optimal discrete Sobolev constant, and show that it is not attained. In the supercritical regime, there exists ${\epsilon }_0={\epsilon }_0(N)>0$ such that $C_{N,\alpha }$ is attained for every $2^{\ast }<\alpha <2^{\ast }+{\epsilon }_0$, with a nonnegative Schwarz symmetric extremal function. The main compactness difficulties stem from the lack of a suitable scaling on ${\mathbb{Z}}^N$ and from the nonhomogeneity of the Born–Infeld energy. We overcome them by combining discrete Schwarz rearrangement with a $Q_1$ discrete-to-continuum comparison and by establishing a strict separation as the ${\mathrm{\ell }}^{\alpha }$-norm tends to infinity.