Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well

Zhang, Fubao; Du, Miao

doi:10.1016/j.jde.2020.07.013

Cited by 50 publications

(16 citation statements)

References 31 publications

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“…Hence, the corresponding results in [9] have been obtained by using the variational techniques in a standard way. In [11][12][13], the authors considered Kirchhoff type problem (3) with a steep potential well. Precisely, the potential function satisfies the following conditions besides (V 2 ):…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the Existence of Solutions for a Class of Schrödinger–Kirchhoff‐Type Equations with Sign‐Changing Potential

Yang

Zhou

Duan

2022

Discrete Dynamics in Nature and Society

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In this paper, we consider the following Kirchhoff problem − a + b ∫ R 3 ∇ u 2 d x Δ u + λ V x u = u p − 2 u , in R 3 u ∈ H 1 R 3 where a , b > 0 are constants, λ is a positive parameter, and 4 < p < 6 . Under suitable assumptions on V x , the existence of nontrivial solution is obtained via variational methods. The potential V x is allowed to be sign-changing.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the Existence of Solutions for a Class of Schrödinger–Kirchhoff‐Type Equations with Sign‐Changing Potential

Yang

Zhou

Duan

2022

Discrete Dynamics in Nature and Society

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“…There are many studies on the solutions of Kirchhoff's equations. [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] We study the fractional Kirchhoff equation with steep potential well. Many studies have investigated the fractional Schrödinger equation, [17][18][19][20][21][22] and recent studies focused on the Kirchhoff type of problem involving the steep potential well.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Multiplicity and concentration of nontrivial solutions for a class of fractional Kirchhoff equations with steep potential well

Shao

Chen

2021

Math Methods in App Sciences

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“…It is worth noting that the potential µV, first introduced by Bartsch and Wang [6], is usually called the steep potential well whose depth is controlled by the parameter µ. Later, the corresponding results were further extended and improved by Jia and Luo [21] and Zhang and Du [35].…”

Section: Introductionmentioning

confidence: 99%

On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms

Sun

Wang

2020

Preprint

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We investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows:When N = 3 and 4 < p < 6, for each a > 0 and µ sufficiently large, we obtain that at least one positive solution exists for 0 < λ ≤ λ 1 (f Ω ) while at least two positive solutions exist for λ 1 (f Ω ) < λ < λ 1 (f Ω ) + δ a without any assumption on the integral Ω g(x)φ p 1 dx, where λ 1 (f Ω ) > 0 is the principal eigenvalue of −∆ in H 1 0 (Ω) with weight function f Ω := f | Ω , and φ 1 > 0 is the corresponding principal eigenfunction. When N ≥ 3 and 2 < p < min{4, 2 * }, for µ sufficiently large, we conclude that (i) at least two positive solutions exist for a > 0 small and 0 < λ < λ 1 (f Ω ); (ii) under the classical assumption Ω g(x)φ p 1 dx < 0, at least three positive solutions exist for a > 0 small and λ 1 (f Ω ) ≤ λ < λ 1 (f Ω ) + δ a ; (iii) under the assumption Ω g(x)φ p 1 dx > 0, at least two positive solutions exist for a > a 0 (p) and λ + a < λ < λ 1 (f Ω ) for some a 0 (p) > 0 and λ + a ≥ 0.

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Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well

Cited by 50 publications

References 31 publications

On the Existence of Solutions for a Class of Schrödinger–Kirchhoff‐Type Equations with Sign‐Changing Potential

On the Existence of Solutions for a Class of Schrödinger–Kirchhoff‐Type Equations with Sign‐Changing Potential

Multiplicity and concentration of nontrivial solutions for a class of fractional Kirchhoff equations with steep potential well

On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms

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