Normalized solutions for nonlinear Schr&amp;ouml;dinger equations

Li, Houwang; Yang, Zuo; Zou, Wenming

doi:10.1360/ssm-2020-0120

Cited by 4 publications

(3 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Condition (2) is called the normalization condition, which imposes a normalization on the L 2 -masses of u and v. The solutions to the system (1) under the constraint (2) are usually referred as normalized solutions. In order to obtain the solution to the system (1) satisfying the normalization condition (2), one need to consider the critical point with the H a (see (4)). Then λ 1 and λ 2 appear as Lagrange multipliers with respect to the mass constraint, which cannot be determined a priori, but are part of the unknown.…”

Section: Introductionmentioning

confidence: 99%

“…The normalized solutions of nonlinear Schrödinger equations and systems have gradually attracted the attention of a large number of researchers in recent years, both for the pure mathematical research and in view of its very important applications in many physical problems; see for more details [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

“…On the one hand, for the nonlinear Schrödinger equation with s = 1, in [5], the author studied existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities p, q which satisfy 2 < q ≤ 2 + 4 N ≤ p, p = q. In [6], the author studied existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities q, 2 * .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Normalized solutions for a coupled fractional Schrödinger system in low dimensions

et al. 2020

Bound Value Probl

View full text Add to dashboard Cite

We consider the following coupled fractional Schrödinger system: $$ \textstyle\begin{cases} (-\Delta )^{s}u+\lambda _{1}u=\mu _{1} \vert u \vert ^{2p-2}u+ \beta \vert v \vert ^{p} \vert u \vert ^{p-2}u, \\ (-\Delta )^{s}v+\lambda _{2}v=\mu _{2} \vert v \vert ^{2p-2}v+\beta \vert u \vert ^{p} \vert v \vert ^{p-2}v \end{cases}\displaystyle \quad \text{in } {\mathbb{R}^{N}}, $$ { ( − Δ ) s u + λ 1 u = μ 1 | u | 2 p − 2 u + β | v | p | u | p − 2 u , ( − Δ ) s v + λ 2 v = μ 2 | v | 2 p − 2 v + β | u | p | v | p − 2 v in R N , with $0< s<1$ 0 < s < 1 , $2s< N\le 4s$ 2 s < N ≤ 4 s and $1+\frac{2s}{N}< p<\frac{N}{N-2s}$ 1 + 2 s N < p < N N − 2 s , under the following constraint: $$ \int _{\mathbb{R}^{N}} \vert u \vert ^{2}\,dx=a_{1}^{2} \quad \text{and}\quad \int _{ \mathbb{R}^{N}} \vert v \vert ^{2}\,dx=a_{2}^{2}. $$ ∫ R N | u | 2 d x = a 1 2 and ∫ R N | v | 2 d x = a 2 2 . Assuming that the parameters $\mu _{1}$ μ 1 , $\mu _{2}$ μ 2 , $a_{1}$ a 1 , $a_{2}$ a 2 are fixed quantities, we prove the existence of normalized solution for different ranges of the coupling parameter $\beta >0$ β > 0 .

show abstract