Theory and Applications of Singular Perturbations by W. Eckhaus, E.M. de Jager

By W. Eckhaus, E.M. de Jager

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Clearly, this theorem shows that there is no essential difficulty in dealing with local maxima of Γ instead of minima. Moreover, when we deal with local minima (resp. maxima) of Γ, the preceding results can be improved because the number of positive solutions of (1) can be estimated by means of the category and M does not need to be a manifold. 9. Let (V-V1) and (J-J1) hold and suppose Γ has a compact set X where Γ achieves a strict local minimum (resp. maximum), in the sense that there exists δ > 0 and a δ-neighborhood Xδ of X such that b := inf{Γ(x) : x ∈ ∂Xδ } > a := Γ|X , (resp.

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If lim inf n→∞ dist(¯ yn , Λ) > 0 then, since we ˜ sufficiently small, we have also lim inf n→∞ dist(¯ ˜ > 0 and so can take Λ yn , Λ) from (46) we get |tn vn |2 = o(1) = ˜ RN \ε−1 n Λ |wn |2 = ˜ RN \ε−1 n Λ |wn (· + zn )|2 = = ˜ yn ) RN \ε−1 n (Λ−¯ |w∞ |2 + o(1), RN which contradicts the positivity of w∞ . Hence we may assume that y¯n → y¯ ∈ Λ. But then β(un ) = RN χ(εn x + y¯n )|wn (x + zn )|2 dx → y¯ ∈ Λ, |wn (x + zn )|2 dx RN ˜ against our (absurd) assumptions β(un ) ∈ / Λ. 4, the map ξ → Φε (ξ) sends M ε ˜ ˜ Then the map into N .

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