On Ambrosetti–Malchiodi–Ni Conjecture for General Hypersurfaces

Wang, Liping; Wei, Juncheng; Yang, Jun

doi:10.1080/03605302.2011.580033

Cited by 16 publications

(23 citation statements)

References 24 publications

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“…Naturally, problem (1.9) is equivalent to problem (1.3) for V (x) = W (εx). It is known that as ε goes to zero, highly concentrated solutions near critical points of the potential W can be found, see [1,10,11] [17]- [20], [25,27,37,42], or near higher dimensional stationary sets of other auxiliary potentials [3,21,33,41]. The number of solutions of (1.9) may depend on the number or type of the critical points of V (x).…”

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confidence: 99%

Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials

2014

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Abstract. We consider the standing-wave problem for a nonlinear Schrödinger equation, corresponding to the semilinear elliptic problemwhere V (x) is a uniformly positive potential and p > 1. Assuming thatfor instance if p > 2, m > 2 and σ > 1 we prove the existence of infinitely many positive solutions. If V (x) is radially symmetric, this result was proved in [43]. The proof without symmetries is much more difficult, and for that we develop a new intermediate Lyapunov-Schmidt reduction method, which is a compromise between the finite and infinite dimensional versions of it.

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confidence: 99%

Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials

2014

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“…As a corollary, there is a constant γ 0 > 0 such that (3.10) The idea for introducing eZ in (3.11) comes directly from [8,33]. The reason is the linear theory in Section 4.2.2, especially Lemma 4.3.…”

Section: Construction Of Approximate Solutionsmentioning

confidence: 99%

“…We present it here in a rather simple and synthetic way since it uses many ideas which have been developed by all the different authors working on this subject or on closely related problems. In particular, we are benefited from the ideas and tricks in [8,31,33].…”

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confidence: 99%

“…In particular they established the validity of the above conjecture for the case n ≥ 2 arbitrary and k = 1. Recently, by applying the method developed in [8], Wang-Wei-Yang [33] considered the one-codimensional case n ≥ 3 and k = n − 1 in the flat Euclidean space R n . The main purpose of this paper is to prove the validity of the above conjecture for all k = 1, .…”

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On the Ambrosetti–Malchiodi–Ni conjecture for general submanifolds

Mahmoudi

Sánchez

Yao

2015

Journal of Differential Equations

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Artículo de publicación ISIWe 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

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“…The method in [16] seems also work for the construction of concentration on hypersurface of higher dimensional case. The reader can also refer to the paper [36] on Ambrosetti-Malchiodi-Ni conjecture for concentration on general hypersurfaces of an inhomogeneous Schrödinger equation.…”

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confidence: 99%

Layered Solutions With Concentration on Lines in Three-Dimensional Domains

Yang

2014

Anal. Appl.

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We consider the following singularly perturbed elliptic problem ε 2 ∆u − u + u p = 0, u > 0 in Ω, ∂u ∂ν = 0 on ∂Ω,where Ω is a bounded domain in R 3 with smooth boundary, ε is a small parameter, 1 < p < ∞, ν is the outward normal of ∂Ω. We employ techniques already developed in [39] to extend their result to three-dimensional domain. More precisely, let Γ be a straight line intersecting orthogonally with ∂Ω at exactly two points and satisfying a non-degenerate condition. We establish the existence of a solution uε concentrating along a curveΓ near Γ, exponentially small in ε at any positive distance from the curve, provided ε is small and away from certain critical numbers. The concentrating curveΓ will collapse to Γ as ε → 0.Problem (1.1) is known as the stationary equation of the Keller-Segel system in chemotaxes [21]. It can also be viewed as a limiting stationary equation for the Geirer-Meinhardt system in biological pattern formation [12]. Even though simplelooking, problem (1.1) has a rich and interesting structure of solutions. For the last 15 years, it has received considerable attention. In particular, various concentration phenomena exhibited by the solutions of (1.1) seem both mathematically intriguing and scientifically useful. We refer to three survey articles [30,31,37] for backgrounds and references.In the pioneering papers [32,33], Ni and Takagi proved the existence of least energy solutions to (1.1), that is, a solution u with minimal energy. Furthermore, they showed in [32,33] that, for each > 0 sufficiently small, u has a spike at the most curved part of the boundary, i.e. the region where the mean curvature attains maximum value.Since the publication of [32, 33], problem (1.1) has received a great deal of attention and significant progress has been made. More specifically, solutions with multiple boundary peaks as well as multiple interior peaks have been established. (13)(14)(15)19,20,34,38] and the references therein.) In particular, it was established in Gui and Wei [15] that for any two given integers k ≥ 0, l ≥ 0 and k + l > 0, problem (1.1) has a solution with exactly k interior spikes and l boundary spikes for every ε sufficiently small. Furthermore, Lin, Ni and Wei [22] showed that there are at least C N (ε|log ε|) −N number of interior spikes. Recently, the first author, Wei and Zeng [2] improve the upper bound to O(ε −N ) which is optimal.It seems natural to ask if problem (1.1) has solutions which "concentrate" on higher dimensional sets, e.g., curves, or surfaces. In this regard, we mention that it has been conjectured for a long time that problem (1.1) actually possesses solutions which have k-dimensional concentration sets for every 0 ≤ k ≤ N − 1. This is often called Ni's conjecture, see [30] or [31]. Progress in this direction, although still limited, has also been made in [26][27][28][29]. In particular, we mention the results of Malchiodi and Montenegro [27, 28] on the existence of solutions concentrating on the whole boundary provided that the sequence ε satisfies some gap c...

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On Ambrosetti–Malchiodi–Ni Conjecture for General Hypersurfaces

Cited by 16 publications

References 24 publications

Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials

Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials

On the Ambrosetti–Malchiodi–Ni conjecture for general submanifolds

Layered Solutions With Concentration on Lines in Three-Dimensional Domains

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