Multiple positive solutions for a coupled nonlinear Hartree type equations with perturbations

Wang, Jun; Dong, Yangyang; He, Qing; Xiao, Lu

doi:10.1016/j.jmaa.2017.01.059

Cited by 10 publications

(3 citation statements)

References 31 publications

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“…Wang et. al in [62] studied the problem (P ) in the local case s = 1 and obtained a partial multiplicity results. We improved their results and showed the multiplicity results with a weaker assumption (3.1) of f 1 and f 2 below.…”

Section: Doubly Nonlocal P-fractional Coupled Elliptic Systemmentioning

confidence: 99%

Critical growth elliptic problems with Choquard type nonlinearity:A survey

Mukherjee¹,

Sreenadh²

2018

Preprint

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This article deals with a survey of recent developments and results on Choquard equations where we focus on the existence and multiplicity of solutions of the partial differential equations which involve the nonlinearity of convolution type. Because of its nature, these equations are categorized under the nonlocal problems. We give a brief survey on the work already done in this regard following which we illustrate the problems we have addressed. Seeking the help of variational methods and asymptotic estimates, we prove our main results.

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Section: Doubly Nonlocal P-fractional Coupled Elliptic Systemmentioning

confidence: 99%

Critical growth elliptic problems with Choquard type nonlinearity:A survey

Mukherjee¹,

Sreenadh²

2018

Preprint

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“…In [22], Wang and Yang established the existence and nonexistence of normalized solutions of system (1.2) with trapping potentials. In [20], Wang obtained the multiplicity of nontrivial solutions of a nonlinearly coupled Choquard system with general subcritical exponents and perturbations. For a Choquard system with upper critical exponents, You, Wang, and Zhao [25,26] derived the existence of a positive ground state of the following system: 5) where N ≥ 5, is a bounded smooth domain in R N , -λ 1 ( ) < λ 1 , λ 2 < 0, and λ 1 ( ) represents the first eigenvalue of -on with the Dirichlet boundary condition.…”

Section: Introductionmentioning

confidence: 99%

Positive ground states for nonlinearly coupled Choquard type equations with lower critical exponents

2021

Bound Value Probl

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We study the coupled Choquard type system with lower critical exponents $$ \textstyle\begin{cases} -\Delta u+\lambda _{1}(x)u=\mu _{1}(I_{\alpha }* \vert u \vert ^{ \frac{N+\alpha }{N}}) \vert u \vert ^{\frac{\alpha }{N}-1}u+\beta (I_{\alpha }* \vert v \vert ^{ \frac{N+\alpha }{N}}) \vert u \vert ^{\frac{\alpha }{N}-1}u,\quad x\in {\mathbb{R}}^{N}, \\ -\Delta v+\lambda _{2}(x)v=\mu _{2}(I_{\alpha }* \vert v \vert ^{ \frac{N+\alpha }{N}}) \vert v \vert ^{\frac{\alpha }{N}-1}v+\beta (I_{\alpha }* \vert u \vert ^{ \frac{N+\alpha }{N}}) \vert v \vert ^{\frac{\alpha }{N}-1}v,\quad x\in {\mathbb{R}}^{N}, \\ u, v\in H^{1}({\mathbb{R}}^{N}), \end{cases} $$ { − Δ u + λ 1 ( x ) u = μ 1 ( I α ∗ | u | N + α N ) | u | α N − 1 u + β ( I α ∗ | v | N + α N ) | u | α N − 1 u , x ∈ R N , − Δ v + λ 2 ( x ) v = μ 2 ( I α ∗ | v | N + α N ) | v | α N − 1 v + β ( I α ∗ | u | N + α N ) | v | α N − 1 v , x ∈ R N , u , v ∈ H 1 ( R N ) , where $N\ge 3$ N ≥ 3 , $\mu _{1}, \mu _{2}, \beta >0$ μ 1 , μ 2 , β > 0 , and $\lambda _{1}(x)$ λ 1 ( x ) , $\lambda _{2}(x)$ λ 2 ( x ) are nonnegative functions. The existence of at least one positive ground state of this system is proved under certain assumptions on $\lambda _{1}$ λ 1 , $\lambda _{2}$ λ 2 .

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“…Recently, Wang and Shi [30] studied the existence and various qualitative properties of positive least energy solutions to system (1.4) with N = 3, α = 1. In [31] the authors acquired the existence and multiplicity of nontrivial solutions of (1.4) with perturbations. In [32] the authors studied the existence and nonexistence of L 2 (R N )-normalized solutions of (1.4) with trapping potentials.…”

Section: Introductionmentioning

confidence: 99%

Existence and asymptotic behavior of positive least energy solutions for coupled nonlinear Choquard equations

You,

Zhao,

Wang

2021

ejde

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In this article, we study the coupled nonlinear Schrodinger equations with Choquard type nonlinearities $$\displaylines{ -\Delta u+\nu_1u=\mu_1(\frac{1}{|x|^{\alpha}} *u^2)u +\beta (\frac{1}{|x|^{\alpha}} *v^2)u \quad\hbox{in } \mathbb{R}^{N},\cr -\Delta v+\nu_2v=\mu_2(\frac{1}{|x|^{\alpha}} *v^2)v + \beta (\frac{1}{|x|^{\alpha}} *u^2)v \quad\hbox{in } \mathbb{R}^{N},\cr u,v \geq 0\quad \hbox{in } \mathbb{R}^{N}, \quad u,v \in H^{1}(\mathbb{R}^{N}), }$$ where $\nu_1,\nu_2,\mu_1,\mu_2$ are positive constants, $\beta>0$ is a coupling constant, $N\geq 3$, $\alpha\in(0,N)\cap (0,4)$, and ``*'' is the convolution operator We show that the nonlocal elliptic system has a positive least energy solution for positive small $\beta$ and positive large $\beta$ via variational methods. For the case in which $\nu_1=\nu_2$, $\mu_1\neq\mu_2$, $N=3,4,5$ and $\alpha=N-2$, we prove the uniqueness of positive least energy solutions. Moreover, the asymptotic behaviors of the positive least energy solutions as $\beta\to 0^{+}$ are studied. For more information see https://ejde.math.txstate.edu/Volumes/2021/47/abstr.html

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Multiple positive solutions for a coupled nonlinear Hartree type equations with perturbations

Cited by 10 publications

References 31 publications

Critical growth elliptic problems with Choquard type nonlinearity:A survey

Critical growth elliptic problems with Choquard type nonlinearity:A survey

Positive ground states for nonlinearly coupled Choquard type equations with lower critical exponents

Existence and asymptotic behavior of positive least energy solutions for coupled nonlinear Choquard equations

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