Some remarks on uniqueness of positive solutions to Kirchhoff type equations

Wu, Ke; Zhou, Fen; Gu, Guangze

doi:10.1016/j.aml.2021.107642

Cited by 9 publications

(5 citation statements)

References 14 publications

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“…So a considerable effort has been devoted by many researcher in the whole space. We only focus on the type of Kirchhoff equations similar to Equation (1.1) now, namely, and its variants, where , , a > 0 and are constants, and is subcritical and superlinear near infinity (see, e.g., [11, 13, 17, 18, 20–22, 30, 31, 33, 34, 36]); one usually assumes that , or V ( x ) is periodic, or , or V ( x ) is coercive.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence of positive solutions for Kirchhoff-type problem in exterior domains

Jia¹,

Li²,

Ma³

2023

Proceedings of the Edinburgh Mathematical Society

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We consider the following Kirchhoff-type problem in an unbounded exterior domain $\Omega\subset\mathbb{R}^{3}$ : (*) \begin{align} \left\{ \begin{array}{ll} -\left(a+b\displaystyle{\int}_{\Omega}|\nabla u|^{2}\,{\rm d}x\right)\triangle u+\lambda u=f(u), & x\in\Omega,\\ \\ u=0, & x\in\partial \Omega,\\ \end{array}\right. \end{align} where a > 0, $b\geq0$ , and λ > 0 are constants, $\partial\Omega\neq\emptyset$ , $\mathbb{R}^{3}\backslash\Omega$ is bounded, $u\in H_{0}^{1}(\Omega)$ , and $f\in C^1(\mathbb{R},\mathbb{R})$ is subcritical and superlinear near infinity. Under some mild conditions, we prove that if \begin{equation*}-\Delta u+\lambda u=f(u), \qquad x\in \mathbb R^3 \end{equation*} has only finite number of positive solutions in $H^1(\mathbb R^3)$ and the diameter of the hole $\mathbb R^3\setminus \Omega$ is small enough, then the problem (*) admits a positive solution. Same conclusion holds true if Ω is fixed and λ > 0 is small. To our best knowledge, there is no similar result published in the literature concerning the existence of positive solutions to the above Kirchhoff equation in exterior domains.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence of positive solutions for Kirchhoff-type problem in exterior domains

Jia¹,

Li²,

Ma³

2023

Proceedings of the Edinburgh Mathematical Society

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“…The proof of ( 4) is immediate from (54). Second, we check (5). Dropping the non negative term t 0 Ω ϕ 2 dx and non positive term (52) in (50) gives…”

Section: Proof Of the Existencementioning

confidence: 99%

“…The existence of solutions was studied, after adding some conditions appropriately. At the same time, there are other arguments worth studying, such as uniqueness of solutions [5,6,7], stability [8,9], etc.…”

Section: Introductionmentioning

confidence: 99%

“…The first innovation of this paper is to study a kind of variation-inequality problem under Kirchhoff operator based on variable exponent p(x). Reference [1][2][3][4][5][6][7][8][9][10][11][12][13] studied the initial boundary value problem with Kirchhoff operator. In this paper, we study a class of variational inequality problems with Kirchhoff operator.…”

Section: Introductionmentioning

confidence: 99%

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Study of weak solutions for a p(x)-Kirchhoff type variation-inequality problems arising from financial problems

Li¹,

Bai²

2023

Preprint

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In this paper, we study variation-inequality initial-boundary value problems with variable parameter Kirchhoff operators. We constructed a map suitable for the Leray-Schauder principle in which the existence of solution is proved. The uniqueness and stability of the solution are also analyzed under a smallness condition on the parameter of Kirchhoff operators. Mathematics Subject Classification 2010: 35K99, 97M30.

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“…Because the Kirchhoff-type equation has a wide range of applications in many fields, such as it models several physical and biological systems, it has been widely considered in the last two decades by using variational methods, see [1][2][3][4][5][6][7][8][9][10][11] and references therein. We just introduce several results closely related to Equation (1) here.…”

Section: Introductionmentioning

confidence: 99%

On Kirchhoff-Type Equations with Hardy Potential and Berestycki–Lions Conditions

Yang

Liu

2023

Mathematics

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The purpose of this paper is to investigate the existence and asymptotic properties of solutions to a Kirchhoff-type equation with Hardy potential and Berestycki–Lions conditions. Firstly, we show that the equation has a positive radial ground-state solution uλ by using the Pohozaev manifold. Secondly, we prove that the solution uλn, up to a subsequence, converges to a radial ground-state solution of the corresponding limiting equations as λn→0−. Finally, we provide a brief summary.

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Some remarks on uniqueness of positive solutions to Kirchhoff type equations

Cited by 9 publications

References 14 publications

Existence of positive solutions for Kirchhoff-type problem in exterior domains

Existence of positive solutions for Kirchhoff-type problem in exterior domains

Study of weak solutions for a p(x)-Kirchhoff type variation-inequality problems arising from financial problems

On Kirchhoff-Type Equations with Hardy Potential and Berestycki–Lions Conditions

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