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Additional info for 1973, Year of the Humanoids: An Analysis of the Fall UFO Humanoid Wave

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On the other hand, by a rotation of coordinates we may assume that the x1 , . . , xN −1 axes lie along principal directions corresponding to λ1 , . . , λN −1 at z. So, the Hessian matrix can be described as ⎤ ⎡ 0 λ1 0 · · · ⎢ 0 λ2 · · · 0 ⎥ ⎥ ⎢ D2 G(x) = ⎢ . ⎥. .. .. ⎦ ⎣ .. . 0 0 ··· λN −1 Thus, at z = (0, 0) we have N −1 D2 G(0)y, y E(w, y) dy RN −1 = i=1 RN −1 λi yi2 E(w, y) dy. ´ and E. Medeiros E. M. do O 20 By the definition of the mass moment of inertia we have that the moment of inertia about the yi -axis, i = 1, .

From this, Iλ (tϕ+ ) → −∞ as t → +∞, and thus, setting e = tϕ+ for t large enough, we derive that e > r and Iλ (e) < Iλ (uλ ). 2]. 1. 1, Problem (PA ) has a positive solution at the mountain pass level for all λ > 0, that is, there is wλ ∈ K verifying Iλ (wλ ) = cλ and Iλ (wλ )(v − wλ ) ≥ 0 ∀v ∈ K, where cλ is the mountain pass level of Iλ . Multiplicity of Positive Solutions 33 Proof. 1 with the Mountain Pass Theorem, we have that the mountain pass level cλ associated with Iλ is a critical value, hence there is wλ ∈ K such that Iλ (wλ ) = cλ and Iλ (wλ )(v − wλ ) ≥ 0 ∀v ∈ K.

1) Ω Let us point out that the norm in W 1,2 (Ω) given by |∇u| dx + |u| dx 2 Ω 2 1 2 Ω is equivalent to the norm · . Indeed, as Ω is a subset of RN which lies between two hyperplanes, by Poincar´e inequality a constant k > 0 exists such that |∇u|2 dx + Ω |u|2 dx ≤ k Ω |∇u|2 dx Ω (see [1, p. 159]). Hence, · is equivalent to the classical norm in W01,2 (Ω). For this reason, even if we have a problem with “zero mass” we don’t use the space D1,2 (Ω) but we study (Pf ) in W01,2 (Ω) as in the “positive mass case”, thus simplifying the argument in [3].

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