Concentration on minimal submanifolds for a singularly by Mahmoudi F., Malchiodi A.

By Mahmoudi F., Malchiodi A.

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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. : Concentration at curves for a singularly perturbed Neumann problem in threedimensional domains, GAFA, to appear. , Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math, 15 (2002), 1507-1568. , Montenegro, Multidimensional Boundary-layers for a singularly perturbed Neumann problem, Duke Math.

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For v = vε and λ = µ, we have the following expansion − ∆gε ψ h (y)∂h u2,ε + ε2 z2 (y, ζ) p−1 + ψ h (y)∂h u2,ε + ε2 z2 (y, ζ) − p (u2,ε ) ψ h (y)∂h u2,ε + ε2 z2 (y, ζ) = ε2 µ −∆gε ψ h (y)∂h u2,ε + ε2 z2 (y, ζ) + ψ h (y)∂h u2,ε + ε2 z2 (y, ζ) = ε2 µ ψ h (y) (−∆gε ∂h w0 + ∂h w0 ) + O(ε3 ) = ε2 µpψ h (y)w0p−1 ∂h w0 + O(ε3 ). 31 From (85) we can expand the Laplacian in the last formula as 2 w0 ∂ya Φj0 − ψ h ∆gε (∂h u2,ε ) = −ε2 ∂y2a ya ψ h ∂h w0 − 2ε2 ∂a ψ h ∂jh −∆gε ψ h (y)∂h u2,ε + 2 w0 + O(ε3 ) 4ε2 ζn+1 Haj ∂ya ψ h ∂jh 2 = −ε2 ∂y2a ya ψ h ∂h w0 − 2ε2 ∂a ψ h ∂jh w0 ∂ya Φj0 − ψ h ∂h (∆gε u2,ε ) 2 2 + 4ε2 ζn+1 Haj ∂ya ψ h ∂jh w0 + ε2 ψ h Rihtj ζt ∂ij w0 3 2 − ε2 ψ h Rillh + Riaah − Γba (Ei )Γab (Eh ) ∂i w0 + O(ε3 ).

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