By Ambrosetti A., Malchiodi A., Ni W.-M.

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To get a similar inequality for p < 2 we need a more accurate argument. From (9), (96) and some elementary computations one finds |z ω |p−1 − |z ω + w|p−1 ≤ Cz p−2 w, for |x − r/ε| ≤ 1 (3| log ε| − C), λ0 − λ 1 for some positive constant C. As a consequence we have (97) |z ω |p−1 − |z ω + w|p−1 ≤ Cε3 e[(2−p)λ0 −λ1 ]|x−r/ε| , for |x − r/ε| ≤ 1 (3| log ε| − C). λ0 − λ 1 for |x − r/ε| ≥ 1 (3| log ε| − C). λ0 − λ 1 On the other hand, from (96) one finds (98) |z ω |p−1 − |z ω + w|p−1 ≤ C|w|p−1 ≤ Cε 3 λ1 (p−1) λ0 −λ1 , If λ1 is chosen sufficiently close to λ0 , then (95), (97) and (98) imply again |Iε (z ω + w)[v1 , v2 ] − Iε (z ω )[v1 , v2 ]| ≤ Cε3 v1 v2 for some constant C.

A. M. A. : Homoclinics: Poincar´e-Melnikov type results via a variational approach. Ann. Inst. H. Poincar´e Analyse Non Lin´eaire 15, 233-252 (1998). : Variational perturbative methods and bifurcation of bound states from the essential spectrum. Proc. Royal Soc. Edinburg 128-A, 1131-1161 (1998). : Semiclassical states of nonlinear Schr¨ odinger equations. Arch. Rational Mech. Anal. 140, 285-300 (1997). : Solutions concentrating on spheres to symmetric singularly pertubed problems. C. Rendus Acad.

Let us consider an approximate solution z Σ of (1) which is concentrated near Σ. In analogy with (102), the energy of such a solution (we are not rescaling in this case) could be expressed as E(zΣ ) ∼ εk (103) V θk dσ, Σ where dσ is the volume element of Σ. g. to the paper [33] for the geometric formulas used below. Let X denote a vector field perpendicular at Σ. Then the Leibnitz rule and the classical formula for the variation of the area yields (104) d E(zΣ ) = εk dX Σ d θk V dX d dσ dX dσ + V θk = εk ∇X V θk − V θk H · X dσ, Σ where H denotes the mean-curvature vector of Σ.

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Singularly perturbed elliptic equations with symmetry existence of solutions concentrating on spheres by Ambrosetti A., Malchiodi A., Ni W.-M.


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