Multiplicity of nontrivial solutions for Kirchhoff type equations with zero mass and a critical term

doi:10.1007/s11464-022-1028-8

Front. Math. China

2022, Vol. 17

Issue (5) : 813-828 https://doi.org/10.1007/s11464-022-1028-8

RESEARCH ARTICLE

Multiplicity of nontrivial solutions for Kirchhoff type equations with zero mass and a critical term

Chongqing WEI(

), Anran LI(

)

School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China

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Abstract

In this paper, a class of Kirchhoff type equations in $R N$ ( $N ⩾ 3$ ) with zero mass and a critical term is studied. Under some appropriate conditions, the existence of multiple solutions is obtained by using variational methods and a variant of Symmetric Mountain Pass theorem. The Second Concentration Compactness lemma is used to overcome the lack of compactness in critical problem. Compared to the usual Kirchhoff-type problems, we only require the nonlinearity to satisfy the classical superquadratic condition (Ambrosetti-Rabinowitz condition).

Keywords Kirchhoff type equations with a critical term variational methods Symmetric Mountain Pass theorem Second Concentration Compactness lemma

Corresponding Author(s): Anran LI

Online First Date: 25 December 2022 Issue Date: 28 December 2022

Cite this article:

Chongqing WEI,Anran LI. Multiplicity of nontrivial solutions for Kirchhoff type equations with zero mass and a critical term[J]. Front. Math. China, 2022, 17(5): 813-828.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-022-1028-8
https://academic.hep.com.cn/fmc/EN/Y2022/V17/I5/813

1	C O Alves, F J Corrêa, G M Figueiredo. On a class of nonlocal elliptic problems with critical growth. Differ Equ Appl 2010; 2(3): 409–417
2	C O Alves, M F Giovany. Nonlinear perturbations of a periodic Kirchhoff equation in RN. Nonlinear Anal 2012; 75(5): 2750–2759
3	C O Alves, M A S Souto. Existence of solution for two classes of elliptic problems in RN with zero mass. J Differential Equations 2012; 252(10): 5735–5750
4	A Ambrosetti, P Rabinowitz. Dual variational methods in critical point theory and applications. J Funct Anal 1973; 14(4): 349–381
5	J Chabrowski. Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents. Calc Var Partial Differential Equations 1995; 3(4): 493–512
6	Z Fan, Q Wu. One Kirchhoff equation involving subcritical or critical Sobolev exponents and weigh function. Acta Math Appl Sin 2019; 42(2): 278–288
7	F FaraciC Farkas. On a critical Kirchhoff-type problem. Nonlinear Anal, 2020, 192: 111679 (14pp)
8	M F FurtadoL D de OliveiraJ P P da Silva. Multiple solutions for a Kirchhoff equation with critical growth. Z Angew Math Phys. 2019, 70(1): 11 (15pp)
9	Z Guo. Ground states for Kirchhoff equations without compact condition. J Differential Equations 2015; 259(7): 2884–2902
10	X He, W Zou. Existence and concentration behavior of positive solutions for a Kirchhoff type equation in R3. J Differential Equations 2012; 252(2): 1813–1834
11	J H Jn, X Wu. Infinitely many radial solutions for Kirchhoff-type problems in RN. J Math Anal Appl 2010; 369(2): 564–574
12	G Kirchhoff, Mechanik, Leipzig: Teubner, 1883
13	Y Li, F Li, J Shi. Existence of a positive solution to Kirchhoff type problems without compactness conditions. J Differential Equations 2012; 253(7): 2285–2294
14	Y Li, F Li, J Shi. Existence of positive solutions to Kirchhoff type problems with zero mass. J Math Anal Appl 2014; 410(1): 361–374
15	P L Lions. The concentration-compactness principle in the calculus of variations. The limit case, I. Rev Mat Iberoam 1985; 1: 145–201
16	T F Ma, J E Rivera. Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl Math Lett 2003; 16(2): 243–248
17	A Mao, Z Zhang. Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal 2009; 70(3): 1275–1287
18	D Naimen. Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent. Nonlinear Differ Equ Appl 2014; 21(6): 885–914
19	K Perera, Z Zhang. Nontrivial solutions of Kirchhoff-type problems via the Yang index. J Differential Equations 2006; 221(1): 246–255
20	E A B Silva, M S Xavier. Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents. Ann Inst H Poincaré Anal Non Linéaire 2003; 20(2): 341–358

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