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首页> 外文期刊>Journal of Functional Analysis >On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains: Clustering concentration layers
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On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains: Clustering concentration layers

机译：On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains: Clustering concentration layers

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We consider the problem epsilon(2)div(del(a(y))u) - V(y)u + u(p) = 0, u > 0 in Omega, where Omega is a bounded domain in R-2 with smooth boundary, the exponent pis greater than 1, epsilon > 0 is a small parameter, Vis a uniformly positive smooth potential on (Omega) over bar, and nu denotes the outward normal of partial derivative Omega. For two positive smooth functions a(1)(y), a(2)(y) on (Omega) over bar, the operator del(a(y)) is given by del(a(y))u - (a1(y)partial derivative u/partial derivative y(1), alpha(2)(y)partial derivative u/partial derivative y(2)). (1). Let Gamma subset of (Omega) over bar be a smooth curve intersecting orthogonally with partial derivative Omega at exactly two points and dividing Omega into two parts. Moreover, Gamma is a non-degenerate geodesicembedded in the Riemannian manifold R2with metric V-2 sigma(y)[a(2)(y)dy(1)(2)+ a(1)(y)dy(2)(2)], where sigma = p+1/p- 1- 1/2. By assuming some additional constraints on the functions a(y), V(y) and the curves G, partial derivative Omega, we prove that there exists a sequence of epsilon such that the problem has solutions u(epsilon) with clustering concentration layers directed along Gamma, exponentially small in epsilon at any positive distance from it. (2). If (Gamma) over tilde is a simple closed smooth curve in Omega(not touching the boundary partial derivative Omega), which is also a non-degenerate geodesicembedded in the Riemannian manifold R-2 with metric V-2 sigma(y)[a(2)(y)dy(1)(2)]+ a(1)(y)dy(2)(2)], then a similar result of concentrated solutions is still true. (C) 2021 Elsevier Inc. All rights reserved.

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