A singularly perturbed Kirchhoff problem revisited

Li, Gongbao; Luo, Peng; Peng, Shuangjie; Wang, Chunhua; Xiang, Chang‐Lin

doi:10.1016/j.jde.2019.08.016

Cited by 63 publications

(33 citation statements)

References 39 publications

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“…Under the assumptions of and , Li et al proved that Equation has a single peak solution via the Lyapunov‐Schmidt reduction method. Moreover, they also declared that this solution is unique provided more detailed information of

V false(x false)

around its local minimal point is known (such as satisfies the hypothesis of below).…”

Section: Introductionmentioning

confidence: 99%

“…As reviewed above, the peak solutions obtained in the literature all concentrate around the minimal points of

V false(x false)

, and the peak solutions which concentrate at other type critical points of

V false(x false)

are not involved. Moreover, for the case where

g false(x, u false)

f false(u false)

is in critical growth, up to now, it is still unclear on the existence of multi‐peak solutions for Equations or .…”

Section: Introductionmentioning

confidence: 99%

“…Instead of directly constructing multi‐peak solutions for or , we introduce suitable change of variables and transform into a semilinear elliptic equation (ie,

b = 0

in ) coupled with an algebraic equation about the rescaling parameter. The basic idea is motivated in previous works, where Equation was investigated for the case of

ε = 1

and

g false(x, u false) = g false(u false)

is autonomous in

x

and subcritical growth. Indeed, Azzollini observed that if

u false(x false)

is a solution of , then the pair

((), v false(x false) : = u false(λ x false), λ)

equivalently solves

({, \begin{matrix} left \\ array - Δ v = g (v) in R^{N}, \\ array λ^{2} - b λ^{N - 2} \int_{R^{N}} | \nabla v |^{2} d x - a = 0 . \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

“…By employing the rescaling of , Lu investigated the existence of multiple solution of provided that

g false(u false)

satisfies Berestycki‐Lions–type conditions. Li et al proved the uniqueness of positive solution of for the special case of

g false(x, u false) = - u + false| u {false|}^{p - 2} u

. However, when

g false(x, u false)

depends on

x

the first equation of in general contains the parameter

λ

, this causes that the solution

v

cannot not be independent of

λ

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Multi‐peak solutions of Kirchhoff equations involving subcritical or critical Sobolev exponents

Wang

Zeng

Zhang

2020

Math Methods in App Sciences

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show abstract

V false(x false)

around its local minimal point is known (such as satisfies the hypothesis of below).…”

Section: Introductionmentioning

confidence: 99%

“…As reviewed above, the peak solutions obtained in the literature all concentrate around the minimal points of

V false(x false)

, and the peak solutions which concentrate at other type critical points of

V false(x false)

are not involved. Moreover, for the case where

g false(x, u false)

f false(u false)

is in critical growth, up to now, it is still unclear on the existence of multi‐peak solutions for Equations or .…”

Section: Introductionmentioning

confidence: 99%

“…Instead of directly constructing multi‐peak solutions for or , we introduce suitable change of variables and transform into a semilinear elliptic equation (ie,

b = 0

in ) coupled with an algebraic equation about the rescaling parameter. The basic idea is motivated in previous works, where Equation was investigated for the case of

ε = 1

and

g false(x, u false) = g false(u false)

is autonomous in

x

and subcritical growth. Indeed, Azzollini observed that if

u false(x false)

is a solution of , then the pair

((), v false(x false) : = u false(λ x false), λ)

equivalently solves

({, \begin{matrix} left \\ array - Δ v = g (v) in R^{N}, \\ array λ^{2} - b λ^{N - 2} \int_{R^{N}} | \nabla v |^{2} d x - a = 0 . \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

“…By employing the rescaling of , Lu investigated the existence of multiple solution of provided that

g false(u false)

satisfies Berestycki‐Lions–type conditions. Li et al proved the uniqueness of positive solution of for the special case of

g false(x, u false) = - u + false| u {false|}^{p - 2} u

. However, when

g false(x, u false)

depends on

x

the first equation of in general contains the parameter

λ

, this causes that the solution

v

cannot not be independent of

λ

.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multi‐peak solutions of Kirchhoff equations involving subcritical or critical Sobolev exponents

Wang

Zeng

Zhang

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…The existence and concentration behavior of positive solutions were studied in [8,9]. The uniqueness and nondegeneracy of positive solutions were obtained by Li et al [14] and the references therein. The existence of multipeak solutions was considered in [23].…”

Section: Introductionmentioning

confidence: 99%

Existence of ground state for fractional Kirchhoff equation with $L^{2}$ critical exponents

Han

Zhang

2020

Bound Value Probl

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Existence, uniqueness, and asymptotic behaviors of ground state solutions of Kirchhoff‐type equation with fourth‐order dispersion

Wang,

Liu

2024

Math Methods in App Sciences

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In this paper, we focus on the following Schrödinger–Kirchhoff‐type problem with fourth‐order dispersion: where are constants and . We make use of Nehari manifold technique together with concentration‐compactness principle to prove that the above equation has at least a ground state solution for if , 6, and 7, and for if . Moreover, we also investigate the asymptotic behaviors of ground state solutions when some coefficients tend to zero. Among them, a uniqueness result about ground state solutions is obtained by implicit function theorem, and a blow‐up result is established by Pohozaev identity if dimension .

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A singularly perturbed Kirchhoff problem revisited

Cited by 63 publications

References 39 publications

Multi‐peak solutions of Kirchhoff equations involving subcritical or critical Sobolev exponents

Multi‐peak solutions of Kirchhoff equations involving subcritical or critical Sobolev exponents

Existence of ground state for fractional Kirchhoff equation with $L^{2}$ critical exponents

Existence, uniqueness, and asymptotic behaviors of ground state solutions of Kirchhoff‐type equation with fourth‐order dispersion

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