On a critical Schrödinger system in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si11.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:math> with steep potential wells

Wu, Yuanze; Zou, Wenming

doi:10.1016/j.na.2019.111643

Cited by 3 publications

(2 citation statements)

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“…Recently, system (1.1) defined in R N with N ≥ 4 has begun to catch mathematicians' attention, see [11,15,17,18,26,[36][37][38][39][40]51] and the references therein. In these cases, the cubic nonlinearities and coupled terms are all of critical growth for N = 4 and even super-critical growth for N ≥ 5, with respect to Sobolev critical exponent.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Standing waves for two-component elliptic system with critical growth in $\mathbb{R}^{4}$: the attractive case

Guo¹,

Li²,

Luo³

et al. 2022

Preprint

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In this paper, we consider the following two-component elliptic system with critical growth) are nonnegative potentials and the nonlinear coefficients β, µ j , j = 1, 2, are positive. Here we also assume λ > 0. By variational methods combined with degree theory, we prove some results about the existence and multiplicity of positive solutions under the hypothesis β > max{µ 1 , µ 2 }. These results generalize the results for semilinear Schrödinger equation on half space by Cerami and Passaseo (SIAM J. Math. Anal., 28, 867-885, (1997)) to the above elliptic system, while extending the existence result from Liu and Liu (Calc. Var.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In contrast, there are only very few results for system (1.1) on the whole R 4 . In [51], Wu and Zou first proved the existence of positive ground state solutions for system (1.1) with steep potential wells, and then showed the phenomenon of phase separation of ground state solutions to system (1.1) as β → −∞.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Standing waves for two-component elliptic system with critical growth in $\mathbb{R}^{4}$: the attractive case

Guo¹,

Li²,

Luo³

et al. 2022

Preprint

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A coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent: Existence and multiplicity of high energy positive solutions

Chen,

Guo,

2024

Journal of Differential Equations

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Solutions to the coupled Schrödinger systems with steep potential well and critical exponent

Lv,

Tang

2024

Advanced Nonlinear Studies

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In the present paper, we consider the coupled Schrödinger systems with critical exponent: − Δ u i + λ V i ( x ) + a i u i = ∑ j = 1 d β i j u j 3 u i u i in R 3 , u i ∈ H 1 ( R N ) , i = 1,2 , … , d , $$\begin{cases}-{\Delta}{u}_{i}+\left(\lambda {V}_{i}\left(x\right)+{a}_{i}\right){u}_{i}=\sum _{j=1}^{d}{\beta }_{ij}{\left\vert {u}_{j}\right\vert }^{3}\left\vert {u}_{i}\right\vert {u}_{i}\quad \,\text{in}\,{\mathbb{R}}^{3},\quad \hfill \\ {u}_{i}\in {H}^{1}\left({\mathbb{R}}^{N}\right),\quad i=1,2,\dots ,d,\quad \hfill \end{cases}$$ where d ≥ 2, β ii > 0 for every i, β ij = β ji when i ≠ j, λ > 0 is a parameter and 0 ≤ V i ∈ L loc ∞ R N $0\le {V}_{i}\in {L}_{\text{loc\,}}^{\infty }\left({\mathbb{R}}^{N}\right)$ have a common bottom int  V i − 1 ( 0 ) ${V}_{i}^{-1}\left(0\right)$ composed of ℓ 0 ℓ 0 ≥ 1 ${\ell }_{0}\left({\ell }_{0}\ge 1\right)$ connected components Ω k k = 1 ℓ 0 ${\left\{{{\Omega}}_{k}\right\}}_{k=1}^{{\ell }_{0}}$ , where int  V i − 1 ( 0 ) ${V}_{i}^{-1}\left(0\right)$ is the interior of the zero set V i − 1 ( 0 ) = x ∈ R N ∣ V i ( x ) = 0 ${V}_{i}^{-1}\left(0\right)=\left\{x\in {\mathbb{R}}^{N}\mid {V}_{i}\left(x\right)=0\right\}$ of V i . We study the existence of least energy positive solutions to this system which are trapped near the zero sets int  V i − 1 ( 0 ) ${V}_{i}^{-1}\left(0\right)$ for λ > 0 large for weakly cooperative case β i j > 0 s m a l l $\left({\beta }_{ij}{ >}0 \mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}\right)$ and for purely competitive case β i j ≤ 0 $\left({\beta }_{ij}\le 0\right)$ . Besides, when d = 2, we construct a one-bump fully nontrivial solution which is localised at one prescribed components Ω k k = 1 ℓ ${\left\{{{\Omega}}_{k}\right\}}_{k=1}^{\ell }$ for large λ.

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On a critical Schrödinger system in R4 with steep potential wells

Cited by 3 publications

References 58 publications

Standing waves for two-component elliptic system with critical growth in $\mathbb{R}^{4}$: the attractive case

Standing waves for two-component elliptic system with critical growth in $\mathbb{R}^{4}$: the attractive case

A coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent: Existence and multiplicity of high energy positive solutions

Solutions to the coupled Schrödinger systems with steep potential well and critical exponent

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