Lemma 30 ([Point fields and mixtures]). For one point charge Q>0 at R , the field in Lemma 29 ([Global comparison problem]) satisfies W(x)≥c(Q|x-R|,|x-R|^-4 ). For a single point charge or radial smooth charge of mass Q , its …

For a single source, Newton's theorem and the Euler equation give ρ(z)≤k(Q/|z-R|)^ 3/2 . For every x,R and s>0 , ∫_ B(x,s) dz |x-z|\,|z-R|^ 3/2 ≤Cs^ 1/2 . To see the uniformity in R , apply H\"older with exponent q_1∈(3/2,2) to the second factor and its conjugate q_1'∈(2,3) to the first. The integrals of both radial powers over B(x,s) are bounded by their integrals over centered balls of the same radius. Their powers of s combine to s^ 1/2 . Split K*ρ(x) at s=Q^ -1/3 . The near part is at most CQ^ 3/2 s^ 1/2 by , and the far part at most 2Q/s by Lemma 29 ([Global comparison problem]). Both are CQ^ 4/3 . For a point charge this gives W≥Q/(2d) for d=|x-R|≤cQ^ -1/3 . Outside that sphere set v=λd^ -4 . For sufficiently small universal λ>0 , Δv=12λd^ -6 ≥ v^ 3/2 , and its boundary value is below the preceding lower bound. On the exterior domain, testing the equation for v-W with (v-W-ε)_+ gives a nonpositive gradient square bounded below by zero. This test has compact support since both fields decay. Thus v≤W . Together with the inner estimate this proves . For the mixture F=∑_i(q_i/q)W_i , convexity gives ΔF≥-4πζ_ truncated + F^p ≥-4πζ+ F^p. Compare with the equation for W and test with (F-W-ε)_+ . The support is compact. Near a point charge whose mass was strictly decreased, F-W tends to -∞ , so the test vanishes there. If the masses agree, the singular terms cancel and the difference is locally H^1 . The remaining density terms are locally in L^ 5/3 , so compact H^1 approximation is valid. Monotonicity of the power gives F≤W .