Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities

Li, Houwang; Zou, Wenming

doi:10.1007/s11784-021-00878-w

Cited by 28 publications

(23 citation statements)

References 20 publications

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“…For instance, when one considers the energy functional corresponding to (1.2) under the mass constraint, in the mass supercritical case p, q > 2 + 4 N there are bounded Palais-Smale sequences that do not have a convergent subsequence and converge weakly to 0. We refer the reader to [7,26,27,28,29,30,39,40,43] for scalar equations, [8,9,10,11,14,21,22,31] for systems of two equations, [35] for systems of k equations.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…It is worth to point out that all the papers about normalized solutions of (1.2) mentioned above, except [31], deal with the Sobolev subcritical case 2 < p, q < 2 * . In [31] only the case 2 < p < 2 * = q has been considered.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…where s ν (u, v) is the global maximum point at positive level, and t ν (u, v) is the minimum point on −∞, log (s) has at most two critical points in R by[31, Remark 3.1], hence it has exactly the two critical points t ν (u, v) and s ν…”

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confidence: 99%

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Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems

Bartsch,

Li,

Zou

2022

Preprint

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The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schrödinger system with critical exponent:We prove that a normalized ground state does not exist for ν < 0. When ν > 0 and α + β ≤ 2 + 4 N , we show that the system has a normalized ground state solution for 0 < ν < ν 0 , the constant ν 0 will be explicitly given. In the case α + β > 2 + 4 N we prove the existence of a threshold ν 1 ≥ 0 such that a normalized ground state solution exists for ν > ν 1 , and does not exist for ν < ν 1 . We also give conditions for ν 1 = 0. Finally we obtain the asymptotic behavior of the minimizers as ν → 0 + or ν → +∞.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems

Bartsch,

Li,

Zou

2022

Preprint

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“…Recently, normalized solutions to elliptic PDEs and systems attract much attention of researchers e.g. [15,16,20,21,25,26]. In [26], N. Soave considered the existence of normalized ground states to the following energy (Sobolev) critical Schrödinger equation…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Multiplicity and asymptotics of standing waves for the energy critical half-wave

Luo¹,

Yang²,

Xiaolong³

2021

Preprint

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In this paper, we consider the multiplicity and asymptotics of standing waves with prescribed mass R N u 2 = a 2 to the energy critical half-wavewhere N ≥ 2, a > 0, q ∈ 2, 2 + 2 N , 2 * = 2N N −1 and λ ∈ R appears as a Lagrange multiplier. We show that (0.1) admits a ground state u a and an excited state v a , which are characterised by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Several asymptotic properties of {u a }, {v a } are obtained and it is worth pointing out that we get a precise description of {u a } as a → 0 + without needing any uniqueness condition on the related limit problem. The main contribution of this paper is to extend the main results in J. Bellazzini et al. [Math. Ann. 371 (2018), 707-740] from energy subcritical to energy critical case. Furthermore, these results can be extended to the general fractional nonlinear Schrödinger equation with Sobolev critical exponent, which generalize the work of H. J. Luo-Z. T. Zhang [Calc. Var. Partial Differ. Equ. 59 (2020)] from energy subcritical to energy critical case.

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“…As for the least energy normalized solutions, N. Soave in [48,49], by restraining the energy functional on a smaller manifold, obtained the existence of ground state normalized solutions with g(u) = |u| p−2 u + µ|u| q−2 u. For more results on normalized solutions for scalar equations and systems, we refer to [7,6,2,3,19,20,30,9]. Now we come back to the Modified Nonlinear Schrödinger equation (1.1).…”

Section: Introductionmentioning

confidence: 99%

Quasilinear Schrödinger equations: ground state and infinitely many normalized solutions

Li,

Zou

2021

Preprint

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In the present paper, we study the normalized solutions for the following quasilinear Schrödinger equations:We first consider the mass-supercritical case p > 4 + 4 N , which has not been studied before. By using a perturbation method, we succeed to prove the existence of ground state normalized solutions, and by applying the index theory, we obtain the existence of infinitely many normalized solutions. Then we turn to study the mass-critical case, i.e., p = 4 + 4 N , and obtain some new existence results. Moreover, we also observe a concentration behavior of the ground state solutions.

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Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities

Cited by 28 publications

References 20 publications

Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems

Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems

Multiplicity and asymptotics of standing waves for the energy critical half-wave

Quasilinear Schrödinger equations: ground state and infinitely many normalized solutions

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