Nonexistence of Stable Solutions to Lane--Emden Equations on Graphs with Polynomial Volume Growth

We study stable solutions of the Lane–Emden equation $$-\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$$ on graphs with polynomial volume growth. Due to the lack of a chain rule in the discrete setting, we establish suitable algebraic inequalities which serve as substitutes for the usual differential identities. Combining these inequalities with the stability condition and cutoff arguments, we obtain nonexistence results for stable solutions in a range of exponents analogous to the Joseph–Lundgren range. As an application, we derive a lower bound for the Sobolev best constant in the discrete setting.