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Nonexistence of Stable Solutions to Lane--Emden Equations on Graphs with Polynomial Volume Growth
We study stable solutions of the Lane--Emden equation $$-Δu=|u|^{p-1}u$$ 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 in
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