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

Abstract: In this paper, we study the sharp Sobolev-type inequality associated with the Born--Infeld energy on the lattice graph (\mathbb Z^N), (N\geq3): [ \frac12\sum_{x\in\mathbb Z^N}\sum_{y\sim x} \left(1-\sqrt{1-|\nabla_{xy}u|^2}\right) \geq C_{N,α} \left(\sum_{x\in\mathbb Z^N}|u(x)|^α\right)^{\frac{N}{N+α}}, ] for (u\in D^{1,2}(\mathbb Z^N)\cap\ell^α(\mathbb Z^N) ) satisfying (|\nabla_{xy}u|\leq1) for every (x\sim y), where (C_{N,α}) denotes the optimal constant. We determine the exact positivity threshold and prove that (C_{N,α}>0) if and only if (α\geq2^*:=2N/(N-2)). At the Sobolev critical exponent (α=2^*), we identify the optimal constant as [ C_{N,2^*}=\frac12\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 (\varepsilon_0=\varepsilon_0(N)>0) such that (C_{N,α}) is attained for every (2^<α<2^+\varepsilon_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 (\ell^α)-norm tends to infinity.