For weighted graphs, the divergence Σ n^{2q−1}/μ(B(o,n))^{q−1}=∞ forces every nonnegative solution of −Δu≥u^q to vanish, resolving the Grigor'yan–Sun–Verbitsky volume-growth conjecture.
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Flow Decomposition, Green Testing, and Lane--Emden Inequalities on Weighted Graphs
For weighted graphs, the divergence Σ n^{2q−1}/μ(B(o,n))^{q−1}=∞ forces every nonnegative solution of −Δu≥u^q to vanish, resolving the Grigor'yan–Sun–Verbitsky volume-growth conjecture.