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

Abstract

In this paper, a class of Kirchhoff type equations in {\mathbb{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

References

[1] Alves C O, Corrêa F J, Figueiredo G M. On a class of nonlocal elliptic problems with critical growth. Differ Equ Appl 2010; 2(3): 409–417

[2] Alves C O, Giovany M F. Nonlinear perturbations of a periodic Kirchhoff equation in RN. Nonlinear Anal 2012; 75(5): 2750–2759

[3] Alves C O, Souto M A S. Existence of solution for two classes of elliptic problems in RN with zero mass. J Differential Equations 2012; 252(10): 5735–5750

[4] Ambrosetti A, Rabinowitz P. Dual variational methods in critical point theory and applications. J Funct Anal 1973; 14(4): 349–381

[5] Chabrowski J. 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] Fan Z, Wu Q. One Kirchhoff equation involving subcritical or critical Sobolev exponents and weigh function. Acta Math Appl Sin 2019; 42(2): 278–288

[7] FaraciFFarkas C. On a critical Kirchhoff-type problem. Nonlinear Anal, 2020, 192: 111679 (14pp)

[8] FurtadoM Fde Oliveira L Dda SilvaJ P P. Multiple solutions for a Kirchhoff equation with critical growth. Z Angew Math Phys. 2019, 70(1): 11 (15pp)

[9] Guo Z. Ground states for Kirchhoff equations without compact condition. J Differential Equations 2015; 259(7): 2884–2902

[10] He X, Zou W. Existence and concentration behavior of positive solutions for a Kirchhoff type equation in R3. J Differential Equations 2012; 252(2): 1813–1834

[11] Jn J H, Wu X. Infinitely many radial solutions for Kirchhoff-type problems in RN. J Math Anal Appl 2010; 369(2): 564–574

[12] KirchhoffG, Mechanik, Leipzig: Teubner, 1883

[13] Li Y, Li F, Shi J. Existence of a positive solution to Kirchhoff type problems without compactness conditions. J Differential Equations 2012; 253(7): 2285–2294

[14] Li Y, Li F, Shi J. Existence of positive solutions to Kirchhoff type problems with zero mass. J Math Anal Appl 2014; 410(1): 361–374

[15] Lions P L. The concentration-compactness principle in the calculus of variations. The limit case, I. Rev Mat Iberoam 1985; 1: 145–201

[16] Ma T F, Rivera J E. Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl Math Lett 2003; 16(2): 243–248

[17] Mao A, Zhang Z. Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal 2009; 70(3): 1275–1287

[18] Naimen D. Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent. Nonlinear Differ Equ Appl 2014; 21(6): 885–914

[19] Perera K, Zhang Z. Nontrivial solutions of Kirchhoff-type problems via the Yang index. J Differential Equations 2006; 221(1): 246–255

[20] Silva E A B, Xavier M S. 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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Higher Education Press 2022