Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case

Chen, Zhijie; Zou, Wenming

doi:10.1007/s00526-014-0717-x

Cited by 108 publications

(128 citation statements)

References 33 publications

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“…For system (1.2) in the critical case 2 p = 2 * , we refer the reader to [8,10,12,15,20] for the bounded domain case, where the existence and asymptotic behaviors of positive least energy solutions and sign-changing solutions were well investigated. For the entire space case = R N , if λ 1 , λ 2 > 0, it is easy to see that (1.2) has no solutions via the method of moving planes.…”

Section: Introductionmentioning

confidence: 99%

Removable singularity of positive solutions for a critical elliptic system with isolated singularity

Chen

Lin

2015

Math. Ann.

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We study qualitative properties of positive singular solutions to a twocoupled elliptic system with critical exponents. This system is related to coupled nonlinear Schrödinger equations with critical exponents for nonlinear optics and BoseEinstein condensates. We prove a sharp result on the removability of the same isolated singularity for both two components of the solutions. We also prove the nonexistence of positive solutions with one component bounded near the singularity and the other component unbounded near the singularity. These results will be applied in a subsequent work where the same system in a punctured ball will be studied.

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Section: Introductionmentioning

confidence: 99%

Removable singularity of positive solutions for a critical elliptic system with isolated singularity

Chen

Lin

2015

Math. Ann.

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“…Semilinear Schrödinger system defined in the whole space or on a bounded smooth domain have received extensive attention in recent years. In fact, the Schrödinger system has a gradient structure, so it can be studied by variational methods, such as and the references therein. Here we obtain some Liouville‐type theorems for the quasilinear Schrödinger system.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Liouville‐type results for a class of quasilinear elliptic systems and applications

Dancer

Yang

Zou

2018

J. London Math. Soc.

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In this paper, we study the following quasilinear elliptic systems {−normalΔmu=f1false(u,vfalse)indouble-struckRN,−normalΔmv=f2false(u,vfalse)indouble-struckRN.Here Δm is the well‐known m‐Laplacian operator, m>1. Under appropriate conditions on the functions f1 and f2, we prove some Liouville‐type nonexistence theorems. The Liouville‐type results can be applied to non‐cooperative quasilinear Schrödinger‐type systems, and we obtain a priori estimates, singularity and decay estimates for the non‐negative solutions of the non‐cooperative quasilinear Schrödinger‐type systems.

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“…Define

false(u_{ε}, v_{ε} false) : = false(ϕ U_{ε}, ϕ V_{ε} false) .

By direct computation (e.g. , , , ), we have

\begin{matrix} true \\ right \int_{Ω} {false| \nabla u_{ε} false|}^{2} & left = \int_{{double-struckR}^{N}} {false| \nabla U false|}^{2} + O ((), ε^{N - 2}), \end{matrix}

\begin{matrix} true \\ right \int_{Ω} \frac{u_{ε}^{2^{*} false(s_{1} false)}}{{| x |}^{s_{1}}} & left = \int_{{double-struckR}^{N}} \frac{U^{2^{*} false(s_{1} false)}}{{| x |}^{s_{1}}} + O ((), ε^{N - s_{1}}), \end{matrix}

\begin{matrix} true \\ right \int_{Ω} \frac{u_{ε}^{α} v_{ε}^{β}}{{| x |}^{s_{2}}} & left = \int_{{double-struckR}^{N}} \frac{U^{α} V^{β}}{{| x |}^{s_{2}}} + O ((), ε^{N - s_{2}}), \end{matrix}

\begin{matrix} true \\ right \int_{Ω} \frac{u_{ε}^{p}}{| x} \end{matrix}

…”

Section: Ground Statesmentioning

confidence: 99%

Ground states for a nonhomogeneous elliptic system involving Hardy–Sobolev critical exponents

Guo

2016

Mathematische Nachrichten

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This paper studies the following nonhomogeneous elliptic system involving Hardy–Sobolev critical exponents {Δu+λ1up−1|x|s+μ1u2*false(s1false)−1|x|s1+αγ2*(s2)uα−1vβ|x|s2=0inΩ,Δv+λ2vp−1|x|s+μ2v2*false(s1false)−1|x|s1+βγ2*(s2)uαvβ−1|x|s2=0inΩ,u≥0,v≥0inΩ,u=0,v=0on∂Ω,where λ1,λ2,μ1,μ2,γ>0,21,α+β=2*false(s2false), Ω is a C1 open bounded domain in RN containing the origin, and N≥4. The existence result of positive ground state solution is established.

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Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case

Abstract: We study the following nonlinear Schrödinger system which is related to Bose-Einstein condensate:

Cited by 108 publications

References 33 publications

Removable singularity of positive solutions for a critical elliptic system with isolated singularity

Removable singularity of positive solutions for a critical elliptic system with isolated singularity

Liouville‐type results for a class of quasilinear elliptic systems and applications

Ground states for a nonhomogeneous elliptic system involving Hardy–Sobolev critical exponents

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