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

Abstract

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

Chen Zhengyan,Zhang Jiafeng. 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.

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References

[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 Schr?dinger equations with steep potential well. Commun Contemp Math, 2001, 3(4): 549-569
[3]	Jiang Y, Zhou H S. Schr?dinger-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]	Poho?aev 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: Birkh?user, 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

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

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