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

Pino, Manuel del; Wei, Juncheng; Yao, Wei

doi:10.1007/s00526-014-0756-3

Cited by 40 publications

(26 citation statements)

References 43 publications

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“…This idea was already implemented by J. Wei and S. Yan in [17] for a different problem regarding the nonlinear Schrödinger equation. We observe that for that problem symmetry assumption was removed in [6]. We suspect that the same can be done also for our problem but this issue seems quite delicate and difficult.…”

Section: Introduction and Main Resultsmentioning

confidence: 77%

Embedded tori with prescribed mean curvature

Caldiroli

Musso

2018

Advances in Mathematics

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We construct a sequence of compact, oriented, embedded, two-dimensional surfaces of genus one into Euclidean 3-space with prescribed, almost constant, mean curvature of the form H(X) = 1 + A|X| −γ for |X| large, when A < 0 and γ ∈ (0, 2). Such surfaces are close to sections of unduloids with small neck-size, folded along circumferences centered at the origin and with larger and larger radii. The construction involves a deep study of the corresponding Jacobi operators, an application of the Lyapunov-Schmidt reduction method and some variational argument.

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Section: Introduction and Main Resultsmentioning

confidence: 77%

Embedded tori with prescribed mean curvature

Caldiroli

Musso

2018

Advances in Mathematics

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“…Before we end the introduction, we want to say a few words on the circulant matrices, which play an important role in this paper. It also appeared as an essential point in the recent work [24,40]. As we will see in Theorem 3.2, the constructed solution concentrates on the vertex of the regular polygon, which equally distributed on the circle.…”

Section: Infinitely Many Solutions Of Fractional Schrödinger Equationmentioning

confidence: 79%

“…For the fractional case where 0 < s < 1, Dávila, del Pino, and Wei [18] recently obtained the first result regarding multiple spikes for the corresponding fractional nonlinear Schrödinger equation with 1 < p < N +2s N −2s . Subsequently, Wang and Zhao [47] extended the result of Wei and Yan [50] (which was also achieved by Wei, Peng, and Yang [49] independently ) to the fractional case, under the following assumption: A natural question is to ask whether or not we can obtain the multiplicity result of (1.1) for a potential V (x) without the radially symmetric assumption adopted in the paper of [24] for the s = 1 case . In this paper, we will provide an affirmative answer to this question under the following assumption on the positive potential V (x):…”

Section: Infinitely Many Solutions Of Fractional Schrödinger Equationmentioning

confidence: 93%

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Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials

Ao¹,

Wei²,

Yang³

2017

Discrete &Amp; Continuous Dynamical Systems - A

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We consider the fractional nonlinear Schrödinger equation (−∆) s u + V (x)u = u p in R N , u → 0 as |x| → +∞, where V (x) is a uniformly positive potential and p > 1. Assuming that V (x) = V∞ + a |x| m + O 1 |x| m+σ as |x| → +∞,and p, m, σ, s satisfy certain conditions, we prove the existence of infinitely many positive solutions for N = 2. For s = 1, this corresponds to the multiplicity result given by Del Pino, Wei, and Yao [24] for the classical nonlinear Schrödinger equation.denotes the Fourier transform. The fractional Laplacian (−∆) s u of a function u ∈ H 2s (R 2 ) is defined in terms of its Fourier transform by the relation2010 Mathematics Subject Classification. 35J10, 35J61. Key words and phrases. Schrödinger equations, circulant matrix, fractional laplacian.5561 5562 WEIWEI AO, JUNCHENG WEI AND WEN YANGProblem (1.1) arises from the study of standing waves ϕ(x, t) = u(x)e iEt for the following nonlinear Schrödinger equations:( 1.2) Equation (1.2) was introduced by Laskin [33] as an extension of the classical nonlinear Schrödinger equation for s = 1, where the Brownian motion of the quantum paths is replaced by a Lévy flight. Namely, if the path integral over Brownian trajectories leads to the well-known Schrödinger equation, then the path integral over Lévy trajectories leads to the fractional Shcrödinger equation. Here, ϕ = ϕ(x, t)represents the quantum mechanical probability amplitude for a given unit-mass particle to have position x at time t (the corresponding probability density is |ϕ| 2 ), under a confinement resulting from the potential V . We refer the reader to [33,34,35] for detailed physical discussions and the motivation for equation (1.2). Note that the fractional Schrödinger case exhibits interesting differences from to the classical case. For instance, the energy of a particle of unit mass is proportional to |p| 2s (rather than |p| 2 ; see, e.g., [33]). Furthermore, space/time scaling of the process implies that the fractal dimension of Lévy paths is 2s (unlike in the classical Brownian case, where it is 2).Lévy processes occur widely in physics, chemistry, and biology for instance in high energy Hamiltonians of relativistic theories and the Heisenberg uncertainty principle. See [17,30] for further motivation of the fractional Laplacian in modern physics. Stable Lévy processes that give rise to equations with fractional Laplacians have recently attracted significant research interest, and there are many results in the literature regarding the existence of such solutions, for example in [1,9,8,18,26,44] and references therein.Let us come back to equation (1.1). When s = 1, i.e., (−∆) s reduces to the standard Laplacian −∆, equation (1.1) becomes − ∆u + V (x)u = u p in R 2 , u(x) → 0 as |x| → +∞.(1.3) Equation (1.3) has been extensively studied in the last thirty years. Ifthen one can show that (1.3) has a least energy (ground state) solution by using the concentration compactness principle (cf. [37,38]). However, if (1.4) does not hold, then problem (1.3) may not admit a least energy solu...

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“…After Wei-Yan's work, there are two problems. One is that can we remove the symmetry on potential V , which is almost done by del Pino-Wei-Yao [16] using the so called intermediate reduction method. The other is whether the condition m > 1 is the best.…”

Section: Liping Wang and Chunyi Zhaomentioning

confidence: 99%

Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential

Wang¹,

Zhao²

2017

Discrete &Amp; Continuous Dynamical Systems - A

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In the paper we prove the multiplicity existence of both nonlinear Schrödinger equation and Schrödinger system with slow decaying rate of electric potential at infinity. Namely, for any m, n > 0, the potentials P, Q have the asymptotic behaviorthen Schrödinger equation and Schrödinger system have infinitely many solutions with arbitrarily large energy, which extends the results of [37] for single Schrödinger equation and [30] for Schrödinger system.

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

Cited by 40 publications

References 43 publications

Embedded tori with prescribed mean curvature

Embedded tori with prescribed mean curvature

Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials

Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential

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