$ \mathbb{R}^4 $ 中一类带陡峭位势的临界 Kirchhoff 型方程的基态解

RichHTML

PDF (PC)

RichHTML

PDF (PC)

Ground State Solutions for a Class of Critical Kirchhoff Type Equation in $\mathbb{R}^4$ with Steep Potential Well

Ground State Solutions for a Class of Critical Kirchhoff Type Equation in R4 \mathbb{R}^4 with Steep Potential Well

Ground State Solutions for a Class of Critical Kirchhoff Type Equation in $\mathbb{R}^4$ with Steep Potential Well

Abstract

References 24

Related Articles 15

Metrics

Comments

Recommended 10

Abstract

Cite this article

share this article

References 24

Related Articles 15

Metrics

Comments

Recommended 10

Abstract

Cite this article

share this article

References 24

Related Articles 15

Metrics

Comments

Recommended 10

Abstract:

In this paper, we focus on dealing with a class of critical Kirchhoff type equation

$\left\{\begin{array}{ll}\displaystyle-\left(a+b\int_{\mathbb{R}^4} |\nabla u|^2\mathrm{d}x\right)\Delta u+\lambda V(x)u =|u|^{2}u +f(u) &\text{ in } \mathbb{R}^4,\\u\in H^{1} (\mathbb{R}^4),\end{array}\right.$

where $a,b > 0$ are constants and $\lambda > 0$ . The nonlinear growth of $|u|^{2}u$ reaches the Sobolev critical exponent since $2^{*}= 4$ in dimension 4. Assume that $V$ is the nonnegative continuous potential, which represents a potential well with the bottom $V^{-1}(0)$ and $f \in C(\mathbb{R},\mathbb{R})$ satisfies suitable conditions. By the variational methods, the existence of at least a ground state solution is obtained. Moreover, we study the concentration behavior of the ground state solutions as $\lambda \rightarrow\infty$ and their asymptotic behavior as $b\rightarrow 0$ and $\lambda \rightarrow\infty$ , respectively.

Key words: Kirchhoff type equation, critical growth, variational methods, steep potential well, ground state solutions

CLC Number:

O177.91

Zhengyan Chen,Jiafeng Zhang. Ground State Solutions for a Class of Critical Kirchhoff Type Equation in $\mathbb{R}^4$ with Steep Potential Well[J].Acta mathematica scientia,Series A, 2025, 45(2): 450-464.

Trendmd

[1]	Bartsch T, Wang Z Q. Existence and multiplicity results for superlinear elliptic problems on $\mathbb{R}^{N}$ . Comm Partial Differential Equations, 1995, 20(9/10): 1725-1741
[2]	Bartsch T, Pankov A, Wang Z Q. Nonlinear Schrödinger equations with steep potential well. Commun Contemp Math, 2001, 3(4): 549-569
[3]	Jiang Y, Zhou H S. Schrödinger-Poisson system with steep potential well. J Differential Equations, 2011, 251(3): 582-608
[4]	Sun J, Chu J, Wu T F. Existence and multiplicity of nontrivial solutions for some biharmonic equations with $p$ -Laplacian. J Differential Equations, 2017, 262(2): 945-977
[5]	Sun J, Wu T F. Ground state solutions for an indefinite Kirchhoff type problem with steep potential well. J Differential Equations, 2014, 256(4): 1771-1792
[6]	Alves C O, Figueiredo G M. Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbb{R}^{N}$ . Nonlinear Anal, 2012, 75(5): 2750-2759
[7]	Wang J, Tian L, Xu J, Zhang F. Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J Differential Equations, 2012, 253(7): 2314-2351
[8]	Figueiredo G M. Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument. J Math Anal Appl, 2013, 401(2): 706-713
[9]	Kirchhoff G. Mechanik. Teubner: Leipzig, 1883
[10]	Azzollini A. The elliptic Kirchhoff equation in $\mathbb{R}^N$ perturbed by a local nonlinearity. Differ Integral Equ, 2012, 25(5/6): 543-554
[11]	Bernstein S. Sur une classe d'équations fonctionnelles aux dérivées partielles. Bull Acad Sci URSS Sér Math, 1940, 4(1): 17-26
[12]	Pohožaev S I. A certain class of quasilinear hyperbolic equations. Mat Sb, 1975, 96(138): 152-166
[13]	Lions J L. On some questions in boundary value problems of mathematical physics. North-Holland Math Stud, 1978, 30: 284-346
[14]	He X M, Zou W M. Ground states for nonlinear Kirchhoff equations with critical growth. Ann Mat Pura Appl, 2014, 193(2): 473-500
[15]	Wang J, Tian L, Xu J, Zhang F. Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J Differential Equations, 2012, 253(7): 2314-2351
[16]	Chen C, Kuo Y, Wu T. The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J Differential Equations, 2011, 250(4): 1876-1908
[17]	Huang Y, Liu Z, Wu Y. On Kirchhoff type equations with critical Sobolev exponent. J Math Anal Appl, 2018, 462(1): 483-504
[18]	Naimen D. The critical problem of Kirchhoff type elliptic equations in dimension four. J Differential Equations, 2014, 257(4): 1168-1193
[19]	Luo L P, Tang C L. Existence and concentration of ground state solutions for critical Kirchhoff-type equation with steep potential well. Complex Var Elliptic Equ, 2022, 67(7): 1756-1771
[20]	Zeng L, Huang Y S. A remark on Kirchhoff-type equations in $\mathbb{R}^4$ involving critical growth. Complex Var Elliptic Equ, 2022, 67(4): 789-806
[21]	Willem M. Minimax Theorems. Boston: Birkhäuser, 1996
[22]	Brézis H, Nirenberg L. Positive soluticns of nonlinear elliptic equations involving critical Sobolev exponent. Comm Pure Appl Math, 1983 36(4): 437-477
[23]	Li G B, Ye H Y. Existence of positive solutions for nonlinear Kirchhoff type problems in $\mathbb{R}^3$ with critical Sobolev exponent. Math Meth Appl Sci, 2014, 37(16): 2570-2584
[24]	Brézis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88(3): 486-490

[1]	Liu Haolin, Guo Helin. The Properties of Normalized Solutions for Mass Critical Kirchhoff Type Equations [J]. Acta mathematica scientia,Series A, 2024, 44(5): 1194-1204.
[2]	Sun Xin, Duan Yu. Multiplicity of Solutions for Sublinear Klein-Gordon-Maxwell Systems [J]. Acta mathematica scientia,Series A, 2024, 44(5): 1205-1215.
[3]	Chen Shangjie. Infinitely Many Large Energy Solutions for the Klein-Gordon-Born-Infeld Equation on $\mathbf{R}^3$ with Concave and Convex Nonlinearities [J]. Acta mathematica scientia,Series A, 2024, 44(3): 637-649.
[4]	Duan Xueliang, Wu Xiaofan, Wei Gongming, Yang Haitao. Multiple Positive Solutions of Kirchhoff Type Linearly Coupled System with Critical Exponent [J]. Acta mathematica scientia,Series A, 2024, 44(3): 699-716.
[5]	Fu Peiyuan, Xia Aliang. Multiplicity of High Energy Solutions for a Class of Nonlocal Critical Elliptic System [J]. Acta mathematica scientia,Series A, 2024, 44(1): 101-119.
[6]	Li Yixian,Zhang Zhengjie. The Existence of Ground State Solutions for a Class of Equations Related to Klein-Gordon-Maxwell Systems [J]. Acta mathematica scientia,Series A, 2023, 43(3): 680-690.
[7]	Xiong Chen, Gao Qi. Locally Minimizing Solutions of a Two-component Ginzburg-Landau System [J]. Acta mathematica scientia,Series A, 2023, 43(2): 321-340.
[8]	Ge Bin, Yuan Wenshuo. Existence and Multiplicity of Radial Solutions for Double Phase Problem on the Entire Space $\mathbb{R} ^N$ [J]. Acta mathematica scientia,Series A, 2023, 43(2): 433-446.
[9]	Anran Li,Dandan Fan,Chongqing Wei. Existence and Asymptotic Behaviour of Solutions for Kirchhoff Type Equations with Zero Mass and Critical [J]. Acta mathematica scientia,Series A, 2022, 42(6): 1729-1743.
[10]	Penghui Zhang,Zhiqing Han. Existence of Nontrivial Solutions for Non-autonomous Kirchhoff-type Equations with Critical Growth in ${{\Bbb R}} ^{3}$ [J]. Acta mathematica scientia,Series A, 2022, 42(5): 1424-1432.
[11]	Yu Duan,Xin Sun. Existence of Positive Solutions for Klein-Gordon-Maxwell Systems with an Asymptotically Linear Nonlinearity [J]. Acta mathematica scientia,Series A, 2022, 42(4): 1103-1111.
[12]	Yanan Wang,Kaimin Teng. Ground State Solutions for Quasilinear Schrödinger Equation of Choquard Type [J]. Acta mathematica scientia,Series A, 2022, 42(3): 730-748.
[13]	Xudong Shang,Jihui Zhang. Existence of Positive Ground State Solutions for the Choquard Equation [J]. Acta mathematica scientia,Series A, 2022, 42(3): 749-759.
[14]	Changmu Chu,Lu Meng. Existence of Positive Solutions for Semilinear Elliptic Equation with Variable Exponent [J]. Acta mathematica scientia,Series A, 2021, 41(6): 1779-1790.
[15]	Xianyong Yang,Xianhua Tang,Guangze Gu. Existence and Multiplicity of Solutions for a Fractional Choquard Equation with Critical or Supercritical Growth [J]. Acta mathematica scientia,Series A, 2021, 41(3): 702-722.

Viewed

Full text

	From	local

	Times	54
	Rate	100%

Abstract

Just accepted	Online first	Issue

0	0	29

From	Others	local

Times	28	1
Rate	97%	3%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed

[1]	Ren Chenchen, Yang Sudan. Existence and Uniqueness of Solutions for Sub-Linear Heat Equations with Almost Periodic Coefficients[J]. Acta mathematica scientia,Series A, 2025, 45(1): 31 -43 .
[2]	Fu Yuechen, Ni Mingkang. The Inner Layer of a Class of Singularly Perturbed High-Order Equations with Discontinuous Right-Hand Side[J]. Acta mathematica scientia,Series A, 2024, 44(5): 1153 -1166 .
[3]	Zhang Wei, Chen Keyuan, Wu Yi, Ni Jinbo. Lyapunov-Type Inequalities for Dirichlet Problems of Multi-Term Caputo Fractional Differential Equations[J]. Acta mathematica scientia,Series A, 2024, 44(6): 1433 -1444 .
[4]	Yang Guicheng, Wen Yangzhe, Mao Jing. A Universal Inequality for the Dirichlet Eigenvalue Problem of the Weighted Laplacian and its Application[J]. Acta mathematica scientia,Series A, 2025, 45(1): 54 -73 .
[5]	Anqi Liu,Ting Yu,Changlin Xiang. Research on the Regularity of a Class of Biharmonic Map-Type Partial Differential Equation Systems[J]. Acta mathematica scientia,Series A, 2025, 45(2): 408 -417 .
[6]	Xiaoying Meng,Lu Lu. Existence and Asymptotic Behavior of Solutions for Kirchhoff Equations Involving the Fractional $p$ -Laplacian[J]. Acta mathematica scientia,Series A, 2025, 45(2): 434 -449 .
[7]	Xufei Lu,Changhua Jiao,Jinghua Yang. Affine Semigroup Dynamical Systems on $\mathbb{Z}_p$ [J]. Acta mathematica scientia,Series A, 2025, 45(2): 305 -320 .
[8]	Nizhou Li,Siyi Zhao,Jialing Zhang. Iteration of a Class of Separable Markov Mappings[J]. Acta mathematica scientia,Series A, 2025, 45(2): 321 -333 .
[9]	Yanxue Lin. Dynamical Localization for the CMV Matrices with Verblunsky Coeffcients Defined by the Skew-Shift[J]. Acta mathematica scientia,Series A, 2025, 45(2): 334 -346 .
[10]	Jianan Cui,Shugen Chai. Indirect Boundary Stabilization of Strongly Coupled Variable Coefficient Wave Equations[J]. Acta mathematica scientia,Series A, 2025, 45(2): 389 -407 .

Just accepted

Online first

Just accepted

Online first