一类带临界指数的非齐次Kirchhoff型问题第二个正解的存在性

Abstract

Abstract:

The following nonhomogeneous Kirchhoff type equations with critical exponent

$\begin{eqnarray*}\left\{\begin{array}{ll}-(a+b\int_{\Omega}|\nabla u|^2{\rm d}x)\Delta u=|u|^{4}u+\lambda f(x), & \ \ x \in \Omega, \\u=0, & \ \ x\in\partial\Omega, \end{array}\right. \end{eqnarray*} $

where $\Omega$ is a smooth bounded domain in ${\Bbb R} ^{3}, $ $a, b, \lambda>0$ are parameters and $f\in L^{\frac{6}{5}}(\Omega)$ is nonzero and nonnegative, are considered. By the variational method, the second positive solution is obtained which completes and improves the resulsts of [3] and [5].

Key words: Kirchhoff equation, Critical exponent, Variational method, Positive solution

CLC Number:

O177.25

Lei Ji,Jiafeng Liao. Existence of the Second Positive Solution for a Class of Nonhomogeneous Kirchhoff Type Problems with Critical Exponent[J].Acta mathematica scientia,Series A, 2019, 39(5): 1094-1101.

Trendmd

References 18

1	Alves C O , Corrêa F J S A , Figueiredo G M . On a class of nonlocal elliptic problems with critical growth. Differential Equation and Applications, 2010, 2 (3): 409- 417
2	Ambrosetti A , Rabinowitz P H . Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14 (4): 349- 381 doi: 10.1016/0022-1236(73)90051-7
3	Benmansour S , Bouchekif M . Nonhomogeneous elliptic problems of Kirchhoff type involving critical Sobolev exponents. Electronic Journal of Differential Equations, 2015, 2015 (69): 1- 11
4	Brézis H , Nirenberg L . Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm Pure Appl Math, 1983, 36 (4): 437- 477 doi: 10.1002/cpa.3160360405
5	丁瑶, 廖家锋. 一类带临界指数的非齐次Kirchhoff型问题正解的存在性. 西南师范大学学报:自然科学版, 2015, 40 (12): 17- 21
	Ding Y , Liao J F . Existence of positive solutions for a class of nonhomogeneous Kirchhoff type problems with critical exponent. J Southwest China Normal Univ(Natural Science Edition), 2015, 40 (12): 17- 21
6	Fiscella A , Valdinoci E . A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal, 2014, 94: 156- 170 doi: 10.1016/j.na.2013.08.011
7	Huang Y S , Liu Z , Wu Y Z . On finding solutions of a Kirchhoff type problem. Proc Amer Math Soc, 2016, 144 (7): 3019- 3033
8	Huang Y S , Liu Z , Wu Y Z . On Kirchhoff type equations with critical Sobolev exponent. J Math Anal Appl, 2018, 462 (1): 483- 503 doi: 10.1016/j.jmaa.2018.02.023
9	Lei C Y , Liu G S , Guo L T . Multiple positive solutions for a Kirchhoff type problem with a critical nonlinearity. Nonlinear Anal Real World Appl, 2016, 31: 343- 355 doi: 10.1016/j.nonrwa.2016.01.018
10	Li H Y , Liao J F . Existence and multiplicity of solutions for a superlinear Kirchhoff-type equations with critical Sobolev exponent in ${\Bbb R^{\Bbb N}}$. Comput Math Appl, 2016, 72 (12): 2900- 2907 doi: 10.1016/j.camwa.2016.10.017
11	廖家锋, 李红英. 带Sobolev临界指数的超线性Kirchhoff型方程正解的存在性与多解性. 数学物理学报, 2017, 37 doi: 10.3969/j.issn.1003-3998.2017.06.011
	Liao J F , Li H Y . Existence and multiplicity of positive solutions for the superlinear Kirchhoff-type equations with critical Sobolev exponent. Acta Mathematica Scientia, 2017, 37A (6): 1119- 1124 doi: 10.3969/j.issn.1003-3998.2017.06.011
12	Liao J F , Li H Y , Zhang P . Existence and multiplicity of solutions for a nonlocal problem with critical Sobolev exponent. Comput Math Appl, 2018, 75 (3): 787- 797 doi: 10.1016/j.camwa.2017.10.012
13	Liu X , Sun Y J . Existence of positive solutions for Kirchhoff type problems with critical exponent. J Part Diff Eq, 2012, 25 (2): 85- 96
14	Naimen D . Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent. NoDEA Nonlinear Differential Equations Appl, 2014, 21 (6): 885- 914 doi: 10.1007/s00030-014-0271-4
15	Naimen D . On the Brezis-Nirenberg problem with a Kirchhoff type perturbation. Adv Nonlinear Stud, 2015, 15 (1): 135- 156
16	Rudin W. Real and Complex Analysis. New York:McGraw-Hill, 1966
17	Willem M. Minimax Theorems. Boston:Birkhauser, 1996
18	Xie Q L , Wu X P , Tang C L . Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Commun Pure Appl Anal, 2013, 12 (6): 2773- 2786 doi: 10.3934/cpaa.2013.12.2773

[1]	Jing Zhang. The Asymptotic Behaviors and Phase Separation for a Class of Subcritical Bose-Einstein Condensation System [J]. Acta mathematica scientia,Series A, 2019, 39(3): 441-450.
[2]	Hailong Yuan,Yuping Wang,Yanling Li. Positive Solutions of a Predator-Prey Model with Cross Diffusion [J]. Acta mathematica scientia,Series A, 2019, 39(3): 545-559.
[3]	Zhide Liu,Zhengping Wang. Least Energy Solution for Nonlinear Kirchhoff Type Elliptic Equation [J]. Acta mathematica scientia,Series A, 2019, 39(2): 264-276.
[4]	Xiaoming Peng,Xiaoxiao Zheng,Yadong Shang. Global Attractors for a Fourth Order Pseudo-Parabolic Equation with Fading Memory [J]. Acta mathematica scientia,Series A, 2019, 39(1): 114-124.
[5]	Zhang Shengui. Infinitely Many Solutions for a Bi-Nonlocal Problem Involving p(x)-Laplacian-Like Operator [J]. Acta mathematica scientia,Series A, 2018, 38(3): 514-526.
[6]	Guo Helin, Wang Yunbo. A Remark on a Constrained Variational Problem [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1125-1128.
[7]	Kang Dongsheng, Xiong Ping. Biharmonic Systems Involving Critical Nonlinearities and Different Rellich-Type Potentials [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1105-1118.
[8]	Liao Jiafeng, Li Hongying. Existence and Multiplicity of Positive Solutions for the Superlinear Kirchhoff-Type Equations with Critical Sobolev Exponent [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1119-1124.
[9]	Chen Lizhen, Li Anran, Li Gang. Existence of Infinitely Many Solutions to a Class of Klein-Gordon-Maxwell System with Superlinear and Sublinear Terms [J]. Acta mathematica scientia,Series A, 2017, 37(4): 663-670.
[10]	Hao Xinan, Liu Lishan. Positive Solutions for Nonlocal Boundary Value Problems of Fourth Order Differential Equation with Parameters [J]. Acta mathematica scientia,Series A, 2017, 37(2): 287-298.
[11]	Huang Decheng, Chen Shouxin. New Proof for the Existence of Minimizing Energy Solutions for the Ginzburg-Landau Equations [J]. Acta mathematica scientia,Series A, 2017, 37(2): 299-306.
[12]	Pei Ruichang, Zhang Jihui, Ma Caochuan. Nontrivial Solutions for a Class of Superlinear (p, 2)-Laplacian Dirichlet Problems [J]. Acta mathematica scientia,Series A, 2017, 37(1): 92-101.
[13]	Xu Jia, Wang Yanxia, Dai Guowei. Fast Heteroclinic Solutions for a Difference Equation Related to Fisher-Kolmogorov's Equation [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1145-1156.
[14]	Zhang Jing. Positive Solutions of Perturbation Elliptic Equation Involving Hardy Potential and Critical Sobolev-Hardy Exponent [J]. Acta mathematica scientia,Series A, 2016, 36(3): 500-506.
[15]	Liu Jian, Zhao Zengqin. Variational Approach to Existence of Weak Solutions to Second-Order Singular Impulsive Problems [J]. Acta mathematica scientia,Series A, 2016, 36(3): 521-530.

Existence of the Second Positive Solution for a Class of Nonhomogeneous Kirchhoff Type Problems with Critical Exponent

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 18

Related Articles 15

Metrics

Comments

Recommended 10