On the Ambrosetti–Malchiodi–Ni conjecture for general submanifolds
Author
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Mahmoudi, Fethi
Author
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Subiabre Sánchez, Felipe
Author
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Yao, Wei
Admission date
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2015-08-04T19:23:30Z
Available date
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2015-08-04T19:23:30Z
Publication date
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2015
Cita de ítem
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J. Differential Equations 258 (2015) 243–280
en_US
Identifier
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DOI: 10.1016/j.jde.2014.09.010
Identifier
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https://repositorio.uchile.cl/handle/2250/132361
General note
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Artículo de publicación ISI
en_US
Abstract
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We study positive solutions of the following semilinear equation
epsilon 2 Delta((g) over bar)u - V(z)u + u(p) = o on M,
where (M, (g) over bar) is a compact smooth n-dimensional Riemannian manifold without boundary or the Euclidean space R-n, epsilon is a small positive parameter, p > 1 and V is a uniformly positive smooth potential. Given k = 1,...,n - 1, and 1 < p < n+2-k/n-2-k. Assuming that K is a k-dimensional smooth, embedded compact submanifold of M, which is stationary and non-degenerate with respect to the functional integral(K) Vp+1/P-1-n-k/2 dvol, we prove the existence of a sequence epsilon = epsilon(j) -> 0 and positive solutions u(epsilon) that concentrate along K. This result proves in particular the validity of a conjecture by Ambrosetti et al. [1], extending a recent result by Wang et al. [32], where the one co-dimensional case has been considered. Furthermore, our approach explores a connection between solutions of the nonlinear Schredinger equation and f -minimal submanifolds in manifolds with density.