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

Li, Houwang; Zou, Wenming

doi:10.48550/arxiv.2006.14387

Cited by 2 publications

(4 citation statements)

References 18 publications

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“…Observe that, if in Theorem 1.3 G i (t) = µ i |t| p i /p i for some µ i > 0 and p i ∈ (2 N , 2 * ), i ∈ {1, 2}, then clearly η = 0 in (1.8) and this result was very recently obtained by Li and Zou in [23,Theorem 1.3], again, unlike this paper, by means of the involved topological argument due to Ghoussoub [19], cf. [4-7, 22, 24, 28, 34].…”

Section: Introductionmentioning

confidence: 54%

Least energy solutions to a cooperative system of Schrödinger equations with prescribed $L^2$-bounds: at least $L^2$-critical growth

Mederski¹,

Schino²

2021

Preprint

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We look for least energy solutions to the cooperative systems of coupled Schrödinger equations   with G ≥ 0, where ρ i > 0 is prescribed andOur approach is based on the minimization of the energy functional over a linear combination of the Nehari and Pohožaev constraints intersected with the product of the closed balls in L 2 (R N ) of radii ρ i , which allows to provide general growth assumptions on G and to know in advance the sign of the corresponding Lagrange multipliers. We assume that G has at least L 2 -critical growth at 0 and Sobolev subcritical growth at infinity. The more assumptions we make on G, N , and K, the more can be said about the minimizers of the energy functional. In particular, if K = 2, N ∈ {3, 4}, and G satisfies further assumptions, then u = (u 1 , u 2 ) is normalized, i.e., R N |u i | 2 dx = ρ 2 i for i ∈ {1, 2}. J(u)

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Section: Introductionmentioning

confidence: 54%

Least energy solutions to a cooperative system of Schrödinger equations with prescribed $L^2$-bounds: at least $L^2$-critical growth

Mederski¹,

Schino²

2021

Preprint

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“…Bartsch and Soave [7] proved the existence of infinitely many normalized solutions to (1.4) with µ 1 = µ 2 > 0 and β ≤ −µ 1 by using a suitable minimax argument. For normalized solutions of nonlinear elliptic systems with more general nonlinearities, we refer the reader to [4,15,25,29].…”

Section: Introductionmentioning

confidence: 99%

“…However, much less is known when the L 2 -norms u 2 , v 2 are prescribed. Recently, Li, Zou in [25] and Mederski, Schino in [29] considered normalized solutions to coupled Schrödinger systems with a wide class of nonlinearities including the Sobolev critical terms. Here, we have to emphasize that the system considered by Li and Zou in [25] contains only a single critical nonlinearity, and coupled terms of the systems considered by Mederski and Schino in [29] must be Sobolev subcritical.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Li, Zou in [25] and Mederski, Schino in [29] considered normalized solutions to coupled Schrödinger systems with a wide class of nonlinearities including the Sobolev critical terms. Here, we have to emphasize that the system considered by Li and Zou in [25] contains only a single critical nonlinearity, and coupled terms of the systems considered by Mederski and Schino in [29] must be Sobolev subcritical. In this paper, we study the existence of normalized solutions to problem (1.1) with doubly Sobolev critical exponent involving coupling terms.…”

Section: Introductionmentioning

confidence: 99%

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Positive normalized solutions to nonlinear elliptic systems in $\R^4$ with critical Sobolev exponent

Luo,

Yang,

Zou

2021

Preprint

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In this paper, we consider the existence and asymptotic behavior on mass of the positive solutions to the following system:under the mass constraintwhere a 1 , a 2 are prescribed, µ 1 , µ 2 , β > 0; α 1 , α 2 ∈ R, p ∈ (2, 4) and λ 1 , λ 2 ∈ R appear as Lagrange multipliers. Firstly, we establish a non-existence result for the repulsive interaction case, i.e., α i < 0(i = 1, 2). Then turning to the case of α i > 0(i = 1, 2), if 2 < p < 3, we show that the problem admits a ground state and an excited state, which are characterized respectively by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Moreover, we give a precise asymptotic behavior of these two solutions as (a 1 , a 2 ) → (0, 0) and a 1 ∼ a 2 . This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behavior of the normalized solutions to Schrödinger systems with Sobolev critical exponent. When 3 ≤ p < 4, we prove an existence as well as non-existence (p = 3) results of the ground states, which are characterized by constrained mountain-pass critical points of the corresponding energy functional. Furthermore, precise asymptotic behaviors of the ground states are obtained when the masses of whose two components vanish and cluster to a upper bound (or infinity), respectively.

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

Cited by 2 publications

References 18 publications

Least energy solutions to a cooperative system of Schrödinger equations with prescribed $L^2$-bounds: at least $L^2$-critical growth

Least energy solutions to a cooperative system of Schrödinger equations with prescribed $L^2$-bounds: at least $L^2$-critical growth

Positive normalized solutions to nonlinear elliptic systems in $\R^4$ with critical Sobolev exponent

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