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

Luo, Xiao; Yang, Xiaolong; Zou, Wenming

doi:10.48550/arxiv.2107.08708

Cited by 5 publications

(7 citation statements)

References 34 publications

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“…in R 4 . The rest of proof is similar to that of Lemma 5.5 in [31], and just needs a slight modification.. From (1.17), we have ) is radially symmetric. By [47,Theorem 1.41] or [26,Lemma 3.5], up to a subsequence, there exists σ 1 := σ 1 (a 1 , a 2 ) such that for some ε 0 > 0,…”

Section: Existence Of a Second Critical Point Of Mountain Pass Typementioning

confidence: 77%

“…Theorem 1.7 indicate that problem (1.1) with mass critical lower order perturbation term possesses at least one normalized ground state solution, whose two components both converge to the Aubin-Talanti bubble in related Sobolev space by making appropriate scaling, as the masses of two components vanish. More recently, in [31] jointly with W. Zou, the first and third authors in this present paper considered the equation…”

Section: Moreovermentioning

confidence: 95%

“…) and λ 1 , λ 2 ∈ R appear as Lagrange multipliers. We must point out that compared with [31], the problem (1.1) in R 4 is more delicate due to the uncertainty of sign of the quadratic interaction term and inhomogeneity of the coupling term. Moreover, since system (1.1) with mixed couplings is asymmetric, the permissible range of b 1 , b 2 obtained in Theorem 1.7 may not be optimal, so it is difficult to prove the non-existence of normalized solutions to (1.1) for larger b 1 and b 2 .…”

Section: Moreovermentioning

confidence: 99%

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Normalized solutions for Schrödinger system with quadratic and cubic interactions

Luo¹,

Wei²,

Xiaolong³

et al. 2021

Preprint

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In this paper, we give a complete study on the existence and non-existence of solutions to the following mixed coupled nonlinear Schrödinger systemHere b 1 , b 2 > 0 are prescribed constants, N ≥ 1, µ 1 , µ 2 , ρ > 0, β ∈ R and the frequencies λ 1 , λ 2 are unknown and will appear as Lagrange multipliers. In the one dimension case, the energy functional is bounded from below on the product of L 2 -spheres, normalized ground states exist and are obtained as global minimizers. When N = 2, the energy functional is not always bounded on the product of L 2spheres. We give a classification of the existence and nonexistence of global minimizers. Then under suitable conditions on b 1 and b 2 , we prove the existence of normalized solutions. When N = 3, the energy functional is always unbounded on the product of L 2 -spheres. We show that under suitable conditions on b 1 and b 2 , at least two normalized solutions exist, one is a ground state and the other is an excited state. Furthermore, by refining the upper bound of the ground state energy, we provide a precise mass collapse behavior of the ground state and a precise limit behavior of the excited state as β → 0. Finally, we deal with the high dimensional cases N ≥ 4. Several non-existence results are obtained if β < 0. When N = 4, β > 0, the system is a mass-energy double critical problem, we obtain the existence of a normalized ground state and its synchronized mass collapse behavior. Comparing with the well studied homogeneous case β = 0, our main results indicate that the quadratic interaction term not only enriches the set of solutions to the above Schrödinger system but also leads to a stabilization of the related evolution system.

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Section: Existence Of a Second Critical Point Of Mountain Pass Typementioning

confidence: 77%

Section: Moreovermentioning

confidence: 95%

Section: Moreovermentioning

confidence: 99%

See 1 more Smart Citation

Normalized solutions for Schrödinger system with quadratic and cubic interactions

Luo¹,

Wei²,

Xiaolong³

et al. 2021

Preprint

Self Cite

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“…Recently, the Schrödinger equation with double power form nonlinearity µ|u| q−2 u + |u| p−2 u has been extensively studied due to Soave [28,29], see [2,16,17,20,30,33] for more results. The multiplicity of normalized solutions to the autonomous Schrödinger equation or systems has also been considered extensively at the last years, see [2,3,5,10,12,14,17,18,19,23,24]. As for the existence of normalized solutions to the non-autonomous Schrödinger equation…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Multiplicity and orbital stability of normalized solutions to non-autonomous Schrödinger equation with mixed nonlinearities

Li¹,

Li²,

Zhu³

2022

Preprint

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This paper studies the multiplicity of normalized solutions to the Schrödinger equation with mixed nonlinearitiesan unknown parameter that appears as a Lagrange multiplier, h is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of h when ǫ is small enough. Moreover, the orbital stability of the solutions obtained is analyzed as well. In particular, our results cover the Sobolev critical case p = 2N/(N − 2).

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“…which was investigated by Zou et al [11], they obtained the existence, nonexistence and asymptotic behavior of normalized ground state solutions for system (1.5) in different cases.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities

Zuo

Rădulescu

2022

Anal.Math.Phys.

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In this paper, we investigate the following fractional Sobolev critical Nonlinear Schrödinger coupled systems: $$\begin{aligned} \left\{ \begin{array}{lll} (-\Delta )^{s} u=\mu _{1} u+|u|^{2^{*}_{s}-2}u+\eta _{1}|u|^{p-2}u+\gamma \alpha |u|^{\alpha -2}u|v|^{\beta } ~ \text {in}~ {\mathbb {R}}^{N},\\ (-\Delta )^{s} v=\mu _{2} v+|v|^{2^{*}_{s}-2}v+\eta _{2}|v|^{q-2}v+\gamma \beta |u|^{\alpha }|v|^{\beta -2}v ~~\text {in}~ {\mathbb {R}}^{N},\\ \Vert u\Vert ^{2}_{L^{2}}=m_{1}^{2} ~\text {and}~ \Vert v\Vert ^{2}_{L^{2}}=m_{2}^{2}, \end{array}\right. \end{aligned}$$ ( - Δ ) s u = μ 1 u + | u | 2 s ∗ - 2 u + η 1 | u | p - 2 u + γ α | u | α - 2 u | v | β in R N , ( - Δ ) s v = μ 2 v + | v | 2 s ∗ - 2 v + η 2 | v | q - 2 v + γ β | u | α | v | β - 2 v in R N , ‖ u ‖ L 2 2 = m 1 2 and ‖ v ‖ L 2 2 = m 2 2 , where $$(-\Delta )^{s}$$ ( - Δ ) s is the fractional Laplacian, $$N>2s$$ N > 2 s , $$s\in (0,1)$$ s ∈ ( 0 , 1 ) , $$\mu _{1}, \mu _{2}\in {\mathbb {R}}$$ μ 1 , μ 2 ∈ R are unknown constants, which will appear as Lagrange multipliers, $$2^{*}_{s}$$ 2 s ∗ is the fractional Sobolev critical index, $$\eta _{1}, \eta _{2}, \gamma , m_{1}, m_{2}>0$$ η 1 , η 2 , γ , m 1 , m 2 > 0 , $$\alpha>1, \beta >1$$ α > 1 , β > 1 , $$p, q, \alpha +\beta \in (2+4s/N,2^{*}_{s}]$$ p , q , α + β ∈ ( 2 + 4 s / N , 2 s ∗ ] . Firstly, if $$p, q, \alpha +\beta <2^{*}_{s}$$ p , q , α + β < 2 s ∗ , we obtain the existence of positive normalized solution when $$\gamma $$ γ is big enough. Secondly, if $$p=q=\alpha +\beta =2^{*}_{s}$$ p = q = α + β = 2 s ∗ , we show that nonexistence of positive normalized solution. The main ideas and methods of this paper are scaling transformation, classification discussion and concentration-compactness principle.

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

Cited by 5 publications

References 34 publications

Normalized solutions for Schrödinger system with quadratic and cubic interactions

Normalized solutions for Schrödinger system with quadratic and cubic interactions

Multiplicity and orbital stability of normalized solutions to non-autonomous Schrödinger equation with mixed nonlinearities

Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities

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