A non-local quasi-linear ground state representation and criticality theory

Abstract

We study energy functionals associated with quasi-linear Schrödinger operators on infinite weighted graphs, and develop a ground state representation. Using the representation, we develop a criticality theory, and show characterisations for a Hardy inequality to hold true. As an application, we show a Liouville comparison principle.

One‐dimensional sharp discrete Hardy–Rellich inequalities

“…The proof of [6, Proposition 3] used somehow an equivalent form of (7.3) with 𝑉 ≡ 1 over ℕ, see also [5,Theorem 3.1] for locally summable graphs.…”

“… $$$$\begin{align} \sum _{x\in X} V(x) |\nabla u|^p_p(x)\geqslant - \sum _{x\in X}\frac{{\rm div}{\left[V (\nabla f)_b^{p-1}\right]}}{f^{p-1}}(x) u^{p}(x), \quad \forall \; u \in C_c(X). \end{align}$$$$The proof of [6, Proposition 3] used somehow an equivalent form of () with $$V \equiv 1$$ over $${\mathbb {N}}$$, see also [5, Theorem 3.1] for locally summable graphs.…”