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By Eisenhart L. P.

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We also set αR (x) := α(x/R), τR (x) := ∇α(x/R) and φR (x) := (αR (x))2 ζ(x). 1), RN α2R ω(B∇ζ) · ∇ζ dx RN ω(B∇ζ) · ∇φR dx − 2 = ≤ 0+2 R≤|x|≤2R RN αR ζω(B∇ζ) · ∇αR dx αR |ζ| ω |(B∇ζ) · ∇αR | dx . 3) R≤|x|≤2R α2R ω(B∇ζ) · ∇ζ dx + δ−1 R≤|x|≤2R ωζ 2 (B∇αR ) · ∇αR dx . 2), R≤|x|≤2R ωζ 2 (B∇αR ) · ∇αR dx = R−2 RN ωζ 2 (B∇τR ) · ∇τR dx ≤ C ′ , for a suitable C ′ > 0. 3), (1 − δ) BR ω(B∇ζ) · ∇ζ dx ≤ (1 − δ) RN α2R ω(B∇ζ) · ∇ζ dx ≤ C ′ δ−1 . By sending R → +∞, we thus obtain that RN ω(B∇ζ) · ∇ζ dx < +∞ and therefore lim R→+∞ |x|≥R ω(B∇ζ) · ∇ζ dx = 0 .

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16. 13). Let u possess one-dimensional 32 ALBERTO FARINA, BERARDINO SCIUNZI, AND ENRICO VALDINOCI symmetry. Then, BR Λ2 (|∇u(x)|) − F (u(x)) + cu dx ≤ K RN −1 , for a suitable K ≥ 1. Proof. 32) BR Λ2 (|∇u(x′ )|) − F (u(x′ )) + cu dx BR Λ2 (h′ (ω · x′ )) − F (h(ω · x′ )) + cu dx BR Λ2 (h′ (ω · x′ )) − F (h(ω · x′ )) + cu dx′ dxN = = = (y ′′ ,t,xN )∈(RN−2 ×R×R)∩BR N −1 ≤ const R Λ2 (h′ (t)) − F (h(t)) + cu dy ′′ dt dxN . 15), BR Λ2 (|∇u(x)|) − F (u(x)) + cu dx = ER (u) + cu |BR | ≤ lim ER (ut ) + cu |BR | + const RN −1 t→+∞ = BR Λ2 (|∇u(x′ )|) − F (u(x′ )) + cu dx + const RN −1 .

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