By Mahmoudi F., Malchiodi A.

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L=Cε−k +1 βl ψl . Since J = −∆N K + O(1), for any integer m one finds ≤ (Jm ϕ, ϕ)L2 (K) m N m−1 ≤ ((−∆N ϕ, ϕ)L2 (K;N K) + (ϕ, ϕ)L2 (K;N K) . K ) ϕ, ϕ)L2 (K;N K) + Cm ((−∆K ) Since ϕ = −k 1 2 Cε l=0 β˜l ϕl , from (106) we deduce that m ≤ (J ϕ, ˜ ϕ) ˜ L2 (K;N K) ε−2m ϕ C 2 ≤ (109) 2m k C 2 2 L2 (K;N K) 2m k + O(ε−2(m−1) ) ϕ 1 −k 2 L2 (K;N K) 2 Cε ε −2m + O(ε −2(m−1) β˜l2 . ) l=0 On the other hand, since in the basis (ψl )l , the function ϕ˜ has non zero components only when l ≥ Cε−k , by the Weyl’s asymptotic formula we have also that ∞ m 2 l=Cε−k +1 µl βl ; (110) (Jm ϕ, ˜ ϕ) ˜ L2 (K;N K) ≥ ∞ CC 2m k ε−2m l=Cε−k +1 βl2 .

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105, (1989), 243-266. , On the Schr¨odinger equation and the eigenvalue Problem. Commun. Math. Phys. 88 (1983), 309-318. [33] Li, Y. , On a singularly perturbed equation with Neumann boundary conditions, Comm. Partial Differential Equations 23 (1998), 487-545. , The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math. 51 (1998), 1445-1490. , Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations, 72 (1988), 1-27. : constant mean curvature hypersurfaces condensing along a submanifold, preprint, 2004.