Normalized Solutions for Lower Critical Choquard Equations with Critical Sobolev Perturbation

Yao, Shuai; Chen, Haibo; Rădulescu, Vicenţiu D.; Sun, Juntao

doi:10.1137/21m1463136

Cited by 46 publications

(7 citation statements)

References 29 publications

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“…Later, Choquard rediscovered it as an approximation of Hartree-Fock's theory of single-component plasma [6] on the modeling of an electron trapped in its own hole. For more details and applications, one can refer to previous studies [7][8][9][10][11][12][13][14][15]. For the case V ≡ 1 and 𝑓 (u) = |u| p−2 u, Moroz and Van Schaftingen [7] obtained a solution to (1.3) when 1 + 𝜇 3 < p < 3 + 𝜇, where 1 + 𝜇 3 and 3 + 𝜇 are the lower and upper critical exponents.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Later, Choquard rediscovered it as an approximation of Hartree–Fock's theory of single‐component plasma [6] on the modeling of an electron trapped in its own hole. For more details and applications, one can refer to previous studies [7–15]. For the case

V\equiv 1

and

f(u)&amp;amp;amp;#x0003D;{\left&amp;amp;amp;#x0007C;u\right&amp;amp;amp;#x0007C;}&amp;amp;amp;#x0005E;{p-2}u

, Moroz and Van Schaftingen [7] obtained a solution to () when

1&amp;amp;amp;#x0002B;\frac{\mu }{3}&amp;amp;lt;p&amp;amp;lt;3&amp;amp;amp;#x0002B;\mu

, where

1&amp;amp;amp;#x0002B;\frac{\mu }{3}

and

3&amp;amp;amp;#x0002B;\mu

are the lower and upper critical exponents.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Ground states for nonlinear fractional Schrödinger–Poisson systems with general convolution nonlinearities

Yang

Liu

Chen

2023

Math Methods in App Sciences

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

V\equiv 1

and

f(u)&amp;amp;amp;#x0003D;{\left&amp;amp;amp;#x0007C;u\right&amp;amp;amp;#x0007C;}&amp;amp;amp;#x0005E;{p-2}u

, Moroz and Van Schaftingen [7] obtained a solution to () when

1&amp;amp;amp;#x0002B;\frac{\mu }{3}&amp;amp;lt;p&amp;amp;lt;3&amp;amp;amp;#x0002B;\mu

, where

1&amp;amp;amp;#x0002B;\frac{\mu }{3}

and

3&amp;amp;amp;#x0002B;\mu

are the lower and upper critical exponents.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Ground states for nonlinear fractional Schrödinger–Poisson systems with general convolution nonlinearities

Yang

Liu

Chen

2023

Math Methods in App Sciences

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“…Of course, a great deal of work has focused on the normalized solution of integerorder nonlinear Schrödinger systems [12][13][14][15][16][17][18] or fractional-order single Schrödinger equations [19][20][21][22][23][24][25][26]. In particular, we highlight that Jeanjean and Lu [21] obtained the existence and multiplicity of normalized solutions and the asymptotic behavior of the ground state solution for a class of mass supercritical problems.…”

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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“…Recently, the existence of multiple solutions for Choquard equations with even or odd nonlinearities was established in Cingolani et al [7]. For more progress on the related study of Choquard equation, we may refer to earlier studies [8–20] for details.…”

Section: Introductionmentioning

confidence: 99%

Multibump positive solutions for Choquard equation with double potentials in ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$

Chen

2023

Math Methods in App Sciences

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Normalized Solutions for Lower Critical Choquard Equations with Critical Sobolev Perturbation

Cited by 46 publications

References 29 publications

Ground states for nonlinear fractional Schrödinger–Poisson systems with general convolution nonlinearities

Ground states for nonlinear fractional Schrödinger–Poisson systems with general convolution nonlinearities

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

Multibump positive solutions for Choquard equation with double potentials in ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$

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Normalized Solutions for Lower Critical Choquard Equations with Critical Sobolev Perturbation

Cited by 46 publications

References 29 publications

Ground states for nonlinear fractional Schrödinger–Poisson systems with general convolution nonlinearities

Ground states for nonlinear fractional Schrödinger–Poisson systems with general convolution nonlinearities

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

Multibump positive solutions for Choquard equation with double potentials in ℝ3$$ {\mathrm{\mathbb{R}}}&#x0005E;3 $$

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Multibump positive solutions for Choquard equation with double potentials in ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$