Least energy positive soultions for $d$-coupled Schrödinger systems with critical exponent in dimension three

Liu, Tianhao; Song, You; Zou, Wenming

doi:10.48550/arxiv.2204.00748

Cited by 2 publications

(1 citation statement)

References 36 publications

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“…Starting from the cerebrated work by Brézis and Nirenberg [5], this critical system has received great attention in the past thirty years, in particular for the existence of positive least energy solutions, we refer to [7,8,16,22] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Positive solution for an elliptic system with critical exponent and logarithmic terms: the higher-dimensional cases

Hajaiej,

Liu,

Zou

2024

J. Fixed Point Theory Appl.

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In this paper, we consider the coupled elliptic system with critical exponent and logarithmic terms:whereN −2 is the Sobolev critical exponent. When N ≥ 5, for different ranges of β, λ i , µ i , θ i , i = 1, 2, we obtain existence and nonexistence results of positive solutions via variational methods. The special case N = 4 was studied by the authors in (arXiv:2304.13822). Note that for N ≥ 5, the critical exponent is given by 2p ∈ (2, 4), whereas for N = 4, it is 2p = 4. In the higher dimensional cases N ≥ 5 brings new difficulties, and requires new ideas. Besides, we also study the Brézis-Nirenberg problem with logarithmic perturbationwhere µ > 0, θ < 0, λ ∈ R, and obtain the existence of positive local minima and least energy solution under some certain assumptions.

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Section: Introductionmentioning

confidence: 99%

Positive solution for an elliptic system with critical exponent and logarithmic terms: the higher-dimensional cases

Hajaiej,

Liu,

Zou

2024

J. Fixed Point Theory Appl.

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Positive solution for an elliptic system with critical exponent and logarithmic terms: the higher dimensional cases

Hajaiej,

LIU,

Zou

2023

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In this paper, we consider the coupled elliptic system with critical exponent and logarithmic terms: \begin{equation} \begin{cases} -\Delta u=\lambda_{1}u+ \mu_1|u|^{2p-2}u+\beta |u|^{p-2}|v|^{p}u+\theta_1 u\log u^2, & \quad x\in \Omega,\\ -\Delta v=\lambda_{2}v+ \mu_2|v|^{2p-2}v+\beta |u|^{p}|v|^{p-2}v+\theta_2 v\log v^2, &\quad x\in \Omega,\\ u=v=0, &\quad x \in \partial \Omega, \end{cases} \end{equation} where $\Omega \subset \R^N$ is a bounded smooth domain, $2p=2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent. When $N \geq 5$, for different ranges of $\beta,\lambda_{i},\mu_i,\theta_{i}$, $i=1,2$, we obtain existence and nonexistence results of positive solutions via variational methods. The special case $N=4 $ was studied by the authors in (arXiv:2304.13822). Note that for $N\geq 5$, the critical exponent is given by $2p\in \sbr{2,4}$, whereas for $N=4$, it is $2p=4$. In the higher dimensional cases $N\geq 5$ brings new difficulties, and requires new ideas. Besides, we also study the Br\'ezis-Nirenberg problem with logarithmic perturbation \begin{equation} -\Delta u=\lambda u+\mu |u|^{2p-2}u+\theta u \log u^2 \quad \text{ in }\Omega, \end{equation} where $\mu>0, \theta<0$, $\lambda\in\R$, and obtain the existence of positive local minima and least energy solution under some certain assumptions.

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Least energy positive soultions for $d$-coupled Schrödinger systems with critical exponent in dimension three

Cited by 2 publications

References 36 publications

Positive solution for an elliptic system with critical exponent and logarithmic terms: the higher-dimensional cases

Positive solution for an elliptic system with critical exponent and logarithmic terms: the higher-dimensional cases

Positive solution for an elliptic system with critical exponent and logarithmic terms: the higher dimensional cases

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