Kirchhoff方程单峰解的局部唯一性

数学物理学报 ›› 2020, Vol. 40 ›› Issue (2): 432-440.

Kirchhoff方程单峰解的局部唯一性

许诗敏(),王春花()

^{华中师范大学数学与统计学学院武汉 430079}

收稿日期:2018-09-05 出版日期:2020-04-26 发布日期:2020-05-21
作者简介:许诗敏, E-mail:xushiminxsm@163.com|王春花, E-mail:chunhuawang@mail.ccnu.edu.cn
基金资助:
国家自然科学基金(11671162)

Local Uniqueness of a Single Peak Solution of a Subcritical Kirchhoff Problem in R³

Shimin Xu(),Chunhua Wang()

^{School of Mathematics and Statistics, Central China Normal University, Wuhan 430079}

Received:2018-09-05 Online:2020-04-26 Published:2020-05-21
Supported by:
国家自然科学基金(11671162)

摘要/Abstract

摘要：

该文主要证明了以下非线Kirchhoff问题的单峰解的局部唯一性

$-\Big(\epsilon^2 a+ \epsilon b \int_{\mathbb{R} ^3} |\nabla u|^2{\rm d}x\Big) \Delta u + u=K(x)|u|^{p-1}u, u>0,x \in \mathbb{R} ^3,$

其中ε>0任意小，a，b>0，1 < p < 5，K：$\mathbb{R}^3$→$\mathbb{R}$是连续有界函数.该文主要采用反证法结合局部的Pohozeav恒等式进行证明.

关键词: 非线性Kirchhoff问题, 局部唯一性, Pohozaev恒等式

Abstract:

In this paper, we obtain the local uniqueness of a single peak solution to the following Kirchhoff problem

$-\Big(\epsilon^2 a+ \epsilon b \int_{\mathbb{R} ^3} |\nabla u|^2{\rm d}x\Big) \Delta u + u=K(x)|u|^{p-1}u, u>0,x \in \mathbb{R} ^3,$

for ε>0 sufficiently small, where a, b>0 and 1 < p < 5 are constants, K: $\mathbb{R}^3$→$\mathbb{R}$ isabounded continuous function. We mainly use a contradiction argument developed by Li G, Luo P, Peng S in[20], applying some local pohozaev identities. Our result is totally new for Kirchhoff equations.

Key words: Local uniqueness, Nonlinear Kirchhoff equations, Pohozaev identities

中图分类号:

O175.2

许诗敏,王春花. Kirchhoff方程单峰解的局部唯一性[J]. 数学物理学报, 2020, 40(2): 432-440.

Shimin Xu,Chunhua Wang. Local Uniqueness of a Single Peak Solution of a Subcritical Kirchhoff Problem in R³[J]. Acta mathematica scientia,Series A, 2020, 40(2): 432-440.

参考文献 21

1	Glangetas L . Uniqueness of positive solutions of a nonlinear elliptic equation involving the critical exponent. Nonlinear Anal, 1993, 20, 571- 603 doi: 10.1016/0362-546X(93)90039-U
2	Oh R . The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent. Journal of Functional Analysis, 1990, 89 (1): 1- 52
3	Grossi M . On the number of single-peak solutions of the nonlinear Schrödinger equation. Annales De Linstitut Henri Poincare Non Linear Analysis, 2002, 19 (3): 261- 280 doi: 10.1016/S0294-1449(01)00089-0
4	Cao D , Heinz H P . Uniqueness of positive multi-lump bound states of nonlinear Schrödinger equations. Mathematische Zeitschrift, 2003, 243 (3): 599- 642 doi: 10.1007/s00209-002-0485-8
5	Deng Y , Lin C , Yan S . On the prescribed scalar curvature problem in $\mathbb{R}^N$, local uniqueness and Periodicity. Journal De Mathématiques Pures Et Appliquées, 2015, 104 (6): 1013- 1044 doi: 10.1016/j.matpur.2015.07.003
6	Cao D , Li S , Luo P . Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations. Calculus of Variations and Partial Differential Equations, 2015, 54 (4): 4037- 4063 doi: 10.1007/s00526-015-0930-2
7	Guo Y , Peng S , Yan S . Local uniqueness and periodicity induced by concentration. Proceedings of the London Mathematical Society, 2017, 114, 1005- 1043 doi: 10.1112/plms.12029
8	Peng S , Wang C , Yan S . Construction of solutions via local Pohozaev idnetities. Journal of Functional Analysis, 2018, 274, 2606- 2633 doi: 10.1016/j.jfa.2017.12.008
9	Deng Y , Peng S , Shuai W . Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}^3$. Journal of Functional Analysis, 2015, 269 (11): 3500- 3527 doi: 10.1016/j.jfa.2015.09.012
10	Figueiredo G , Ikoma N , Santos J . S Existence and concentration result for the Kirchhoff type equations with general nonlinearities. Archive for Rational Mechanics and Analysis, 2014, 123 (3): 931- 979
11	Guo Z . Ground states for Kirchhoff equations without compact condition. Journal of Differential Equations, 2015, 259 (7): 2884- 2902 doi: 10.1016/j.jde.2015.04.005
12	He Y , Li G . Standing waves for a class of Kirchhoff type problems in $\mathbb{R}^3$ involving critical Sobolev exponents. Calculus of Variations and Partial Differential Equations, 2015, 54 (3): 3067- 3106 doi: 10.1007/s00526-015-0894-2
13	He X , Zou W . Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Analysis:Theory, Methods and Applications, 2010, 70 (3): 1407- 1414
14	He X , Zou W . Multiplicity of solutions for a class of Kirchhoff type problems. Acta Mathematicae Applicatae Sinica, 2010, 70 (3): 387- 394
15	Li Y , Li F , Shi J . Existence of a positive solution to Kirchhoff type problems without compactness conditions. Journal of Differential Equations, 2012, 253 (7): 2285- 2294 doi: 10.1016/j.jde.2012.05.017
16	Li G , Ye H . Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^3$. Journal of Differential Equations, 2014, 257 (2): 566- 600 doi: 10.1016/j.jde.2014.04.011
17	Lu S . An autonomous Kirchhoff-type equation with general nonlinearity in $\mathbb{R}^N$. Nonlinear Analysis:Real World Applications, 2017, 34, 233- 249 doi: 10.1016/j.nonrwa.2016.09.003
18	Perera K , Zhang Z . Nontrivial solutions of Kirchhoff-type problems via the Yang index. Journal of Differential Equations, 2006, 221 (1): 246- 255 doi: 10.1016/j.jde.2005.03.006
19	Zhang Z , Perera K . Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. Journal of Mathematical Analysis and Applications, 2006, 317 (2): 456- 463 doi: 10.1016/j.jmaa.2005.06.102
20	Li G, Luo P, Peng S, et al. Uniqueness and nondegeneracy of positive solutions to Kirchhoff equations and its applications in singular perturbation problems. 2017, arXiv:1703.05459
21	Cao D , Li S , Luo P . Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations. Calculus of Variations and Partial Differential Equations, 2015, 54 (4): 4037- 4063 doi: 10.1007/s00526-015-0930-2

[1]	胡良根. 非线性Hénon方程解的Liouville定理[J]. 数学物理学报, 2016, 36(4): 639-648.
[2]	杜刚. 一类带临界指标的Neumann问题解的唯一性[J]. 数学物理学报, 2009, 29(3): 810-822.
[3]	沈尧天. 拟线性椭圆型方程组奇性解的 Pohozaev 恒等式与解不存在性[J]. 数学物理学报, 1997, 17(4): 389-395.