Attainment for all supercritical exponents on the lattice

Determine whether the optimal constant for the discrete Born–Infeld Sobolev inequality on the lattice graph \(\mathbb Z^N\) is attained for every exponent \(\alpha>2^*=2N/(N-2)\).

Background

The paper studies the optimal constant $C_{N,\alpha}$ in the discrete Born–Infeld Sobolev inequality on $\mathbb Z^N$. It proves that $C_{N,\alpha}>0$ precisely when $\alpha\geq 2^*$, identifies the critical value $C_{N,2^*}=\frac12\mathcal S_2$, and shows that the critical constant is not attained.

For supercritical exponents, the paper proves attainment only in a near-critical interval $2^*<\alpha<2^*+\varepsilon_0$. Although compactness at each fixed $\ell^\alpha$-constraint level holds for every $\alpha>2^*$, the strict variational gap required to prevent minimizing sequences from escaping to infinity is established only near $2^*$. The unresolved issue is whether attainment extends to every supercritical exponent, as it does in the corresponding continuous problem.