Title: On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains
Abstract: We consider a singularly perturbed elliptic problem on a smooth two dimensional bounded domain.Let $\Gamma$ be a curve intersecting orthogonally with the boundary at exactly two points and dividing the domain into two parts. Moreover, $\Gamma$ satisfies stationary and non-degeneracy conditions with respect to the arc length functional . We prove the existence of a solution concentrating along the whole of $\Gamma$, exponentially small at any positive distance from it, provided that small parameter is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by A. Ambrosetti, A. Malchiodi and W.-M. Ni (p.327, Indiana Univ. Math. J. 53 (2004), no. 2).
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