Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity

Bae, Soohyun; Byeon, Jaeyoung

doi:10.3934/cpaa.2013.12.831

Cited by 9 publications

(11 citation statements)

References 32 publications

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“…Proof. The following proof is similar to [6], and we give the proof for the completeness. For any fixed α > 0, we define Γ ε (x) = 1 |x|(log |x|) α , (3.44) then there exists some C > 0 such that min x∈∂Oε Γ ε (x) ≥ Cε 2 .…”

Section: From (324) and (325) We See That Lim Infmentioning

confidence: 92%

Standing waves for coupled nonlinear Schrödinger equations with decaying potentials

Chen

Zou

2013

Journal of Mathematical Physics

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We study the following singularly perturbed problem for a coupled nonlinear Schrödinger system which arises in Bose-Einstein condensate:Here, a, b are nonnegative continuous potentials, and µ 1 , µ 2 > 0. We consider the case where the coupling constant β > 0 is relatively large. Then for sufficiently small ε > 0, we obtain positive solutions of this system which concentrate around local minima of the potentials as ε → 0. The novelty is that the potentials a and b may vanish at someplace and decay to 0 at infinity.

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Section: From (324) and (325) We See That Lim Infmentioning

confidence: 92%

Standing waves for coupled nonlinear Schrödinger equations with decaying potentials

Chen

Zou

2013

Journal of Mathematical Physics

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“…Then, sinceū converges to 0 at infinity, (11) can be regarded as a perturbation of supersolution for (4) in the exterior domain |x| ≥ R. So, we can apply the arguments in Bae-Byeon [25]. We define v(t) ≡ r mū (r) with t = ln r and m = 2 p−1 .…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Bae and Byeon in Reference [25] found almost optimal threshold on the decaying condition of V(x) at infinity between existence and nonexistence of positive solutions of (4). By the decaying condition, the Equation 3has the threshold for the existence and nonexistence of positive solution.…”

Section: Introductionmentioning

confidence: 99%

Nonexistence of Positive Solutions for Quasilinear Equations with Decaying Potentials

Kwon¹

2020

Mathematics

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“…In Sect. 5, we shall see that the growth restriction on K (x) at infinity can be weaken to the following form: there exist τ <…”

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confidence: 99%

Semiclassical states for nonlinear Dirac equations with singular potentials

Wang

Zhang

2021

Calc. Var.

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In this paper, we study the following nonlinear Dirac equationwhere p ∈ (2, 3), a > 0 is a constant, α = (α 1 , α 2 , α 3 ), α 1 , α 2 , α 3 and β are 4×4 Pauli-Dirac matrices. Our investigation focuses on the case in which V (x) may attain ±a at somewhere or at infinity and K (x) may approach ±∞ as x accumulating at some points and as |x| → ∞. This is a case with the potentials being singular, which has not been studied before as all the works in the literature require lim sup |x|→∞ |V (x)| < a and K ∈ L ∞ (R 3 , R + ). Under some mild assumptions on V and K , for ε > 0 small, we construct localized bound state solutions which concentrate around the singular set of K . Mathematics Subject Classification 35Q40 • 49J35 1 Introduction and main resultsIn this paper, we consider the existence and concentration behavior of semiclassical states of the following stationary Dirac equationCommunicated by M. del Pino.

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Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity

Cited by 9 publications

References 32 publications

Standing waves for coupled nonlinear Schrödinger equations with decaying potentials

Standing waves for coupled nonlinear Schrödinger equations with decaying potentials

Nonexistence of Positive Solutions for Quasilinear Equations with Decaying Potentials

Semiclassical states for nonlinear Dirac equations with singular potentials

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