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

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Loads of financial difficulties can formulated as limited optimizations and equilibration in their ideas. numerous mathematical theories were delivering economists with necessary machineries for those difficulties bobbing up in monetary idea. Conversely, mathematicians were encouraged via quite a few mathematical problems raised via fiscal theories.

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Extra info for Singularly perturbed elliptic equations with symmetry existence of solutions concentrating on spheres

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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.