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

RESUMO

In this paper, we study the sharp Sobolev-type inequality associated with the Born--Infeld energy on the lattice graph \(\mathbb Z^N\), \(N\geq3\): \[ \frac12\sum_{x\in\mathbb Z^N}\sum_{y\sim x} \left(1-\sqrt{1-|\nabla_{xy}u|^2}\right) \geq C_{N,\alpha} \left(\sum_{x\in\mathbb Z^N}|u(x)|^\alpha\right)^{\frac{N}{N+\alpha}}, \] for \(u\in D^{1,2}(\mathbb Z^N)\cap\ell^\alpha(\mathbb Z^N) \) satisfying \(|\nabla_{xy}u|\leq1\) 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 \(\alpha\geq2^*:=2N/(N-2)\). At the Sobolev critical exponent \(\alpha=2^*\), 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 \(C_{N,\alpha}\) is attained for every \(2^*<\alpha<2^*+\varepsilon_0\), 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 \(\ell^\alpha\)-norm tends to infinity.