Concentration behavior of nonlinear Hartree-type equation with almost mass critical exponent

Li, Yuan; Zhao, Dun; Wang, Qingxuan

doi:10.1007/s00033-019-1172-5

Cited by 7 publications

(2 citation statements)

References 36 publications

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“…In particular, Wang and Li 26 obtained a blow‐up profile of boosted ground states for () as N ↗ N c ( v ) with v fixed. Similar blow‐up results appeared in studying nonlinear elliptic equations with L 2 ‐critical nonlinearity, one can see previous studies 30–32 and the references therein.…”

Section: Introduction and Main Resultssupporting

confidence: 81%

A blow‐up result for the travelling waves of the pseudo‐relativistic Hartree equation with small velocity

Wang

2021

Math Methods in App Sciences

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In this paper, we consider the pseudo‐relativistic Hartree equation i∂tψ=−△+m2ψ−1|x|∗|ψ|2ψonℝ3 and study travelling solitary waves of the form ψ(t, x) = eitμφ(x − v t) , where v∈ℝ3 denotes travelling velocity. Fröhlich, Jonsson and Lenzmann in [Comm. Math. Phys. 2007, 274:1‐30] proved that for |v|<1 there exists a critical constant Nc(v), such that the travelling waves exist if and only if 0 < N < Nc(v), where N denotes particle number. In this paper, we consider v=false(β,0,0false) with 0 < β < 1, and let Ncfalse(βfalse)=Ncfalse(vfalse)false|v=false(β,0,0false). We find that Nc(β) is Lipschitz continuity with respect to β. Based on this fact, we then prove that the boosted ground states φβ with false‖φβfalse‖L22=false(1−βfalse)Ncfalse(βfalse) satisfy limβ→0+false‖φβfalse‖H1false/2→+∞. The explicit blow‐up profile and rate will be computed.

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Section: Introduction and Main Resultssupporting

confidence: 81%

A blow‐up result for the travelling waves of the pseudo‐relativistic Hartree equation with small velocity

Wang

2021

Math Methods in App Sciences

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“…Hence, we can assume (after taking subsequence), lim n→∞ u * n − φ L q = 0 for any 2 ≤ q < 6. As a consequence, by the triangle inequality and the Hardy-Littlewood-Sobolev inequality or see [21,Lemma 2.1], we can obtain…”

Section: Existence and Non-degeneracymentioning

confidence: 97%

On mass - critical NLS with local and non-local nonlinearities

Georgiev¹,

Li²

2022

Preprint

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We consider the following nonlinear Schrödinger equation with the double L 2 -critical nonlinearitieswhere µ > 0 is small enough. Our first goal is to prove the existence and the non-degeneracy of the ground state Q µ . In particular, we develop an appropriate perturbation approach to prove the radial non-degeneracy property and then obtain the general non-degeneracy of the ground state Q µ . We then show the existence of finite time blowup solution with minimal mass u 0 L 2 = Q µ L 2 . More precisely, we construct the minimal mass blowup solutions that are parametrized by the energy E µ (u 0 ) > 0 and the momentum P µ (u 0 ). In addition, the nondegeneracy property plays crucial role in this construction.

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Controllability of nonlinear Hilfer fractional Langevin dynamical system

Radhakrishnan

Sathya

2022

Numerical Methods Partial

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The main objective of this article is to prove the controllability result of a nonlinear Hilfer fractional Langevin dynamical system in an abstract-weighted space. This paper initiates with the persistence of the control formula for the Hilfer fractional Langevin dynamical system. Furthermore, As ensue is extended to the controllability result of Hilfer fractional Langevin dynamical system by the use of Krasnoselskii fixed point theorem. Ultimately, an example is furnished to demonstrate our outcomes.

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Concentration behavior of nonlinear Hartree-type equation with almost mass critical exponent

Cited by 7 publications

References 36 publications

A blow‐up result for the travelling waves of the pseudo‐relativistic Hartree equation with small velocity

A blow‐up result for the travelling waves of the pseudo‐relativistic Hartree equation with small velocity

On mass - critical NLS with local and non-local nonlinearities

Controllability of nonlinear Hilfer fractional Langevin dynamical system

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