December  2016, 9(6): 1663-1685. doi: 10.3934/dcdss.2016069

Existence of positive solutions for a class of Kirchhoff type equations in $\mathbb{R}^3$

1. 

School of Mathematics and Computer Science, Hubei University of Arts and Science, Xiangyang, Hubei 441053, China

2. 

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061

Received  July 2015 Revised  September 2016 Published  November 2016

The paper deals with the following equation of Kirchhoff type, \begin{align*} & -\left ( 1+b\left(\int_{\mathbb{R}^3}|\nabla u|^2dx\right)^r\, \right ) \Delta u+u=k(x)\left (|u|^{q-2}u+\theta g(u)\right )+\lambda h(x)u \end{align*} with $x\in \mathbb{R}^3$, where $u\in H^{1}(\mathbb{R}^3)$, $ b > 0, $ $0 < r < 2, q \in [2(r+1), 6) $, $\theta $ is a small constant, $\lambda$ is a parameter, and a weight function $h (x) \geq 0$. It is known that the linear operator $-\Delta u+u-\lambda h(x)u$ is coercive if $0<\lambda<\lambda_1(h)$ and is non-coercive if $\lambda>\lambda_1(h)$, where $\lambda_1(h)$ is the first eigenvalue of the operator $-\Delta u +u $ with the weight $h(x)$. Under suitable conditions on the functions $k(x)$ and $g(s)$, it is shown that the equation has a positive solution for any $\lambda\in(0,\lambda_1(h))$ and two positive solutions for $\lambda\in(\lambda_1(h), \lambda_1(h) + \tilde \delta )$ with $\tilde \delta > 0$ small. The conditions imposed on $k(x) $ and $g(s)$ are much weaker than those used before, thereby generalizing several existing results on the existence of positive solutions for this type of Kirchhoff equations.
Citation: Ling Ding, Shu-Ming Sun. Existence of positive solutions for a class of Kirchhoff type equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1663-1685. doi: 10.3934/dcdss.2016069
References:
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C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,, Comput. Math. Appl., 49 (2005), 85. doi: 10.1016/j.camwa.2005.01.008. Google Scholar

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C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbbR^N $,, Nonlinear Anal., 75 (2012), 2750. doi: 10.1016/j.na.2011.11.017. Google Scholar

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B. T. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems,, J. Math. Anal. Appl., 394 (2012), 488. doi: 10.1016/j.jmaa.2012.04.025. Google Scholar

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S. Cingolani and J. L. Gomez, Positive solutions of a semilinear elliptic equation on $\mathbbR^N$ with indefinite nonlinearity,, Adv. Diff. Eq., 1 (1996), 773. Google Scholar

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D. G. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in $\mathbbR^N$,, Calc. Var. Partial Differential Equations, 13 (2001), 159. doi: 10.1007/PL00009927. Google Scholar

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X. M. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems,, Nonlinear Anal., 70 (2009), 1407. doi: 10.1016/j.na.2008.02.021. Google Scholar

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X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^3$,, J. Differential Equations, 252 (2012), 1813. doi: 10.1016/j.jde.2011.08.035. Google Scholar

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J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$,, J. Math. Anal. Appl., 369 (2010), 564. doi: 10.1016/j.jmaa.2010.03.059. Google Scholar

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G. Kirchhoff, Mechanik,, Teubner, (1883). Google Scholar

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L. Liu and C. S. Chen, Study on existence of solutions for $p$-Kirchhoff elliptic equation in $\mathbbR^N$ with vanishing potential,, Journal of Dynamical and Control Systems, 20 (2014), 575. doi: 10.1007/s10883-014-9244-5. Google Scholar

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W. Liu and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations,, J. Appl. Math. Comput., 39 (2012), 473. doi: 10.1007/s12190-012-0536-1. Google Scholar

[23]

J. L. Lions, On some questions in boundary value problems of mathematical physics,, North-Holland Math. Stud., 30 (1978), 284. Google Scholar

[24]

P. L. Lions, The Concentration-Compactness Principle in the Calculus of Variations. The Locally Compact Case, Part I,, Ann. Inst. H. Poincaré, 1 (1984), 109. Google Scholar

[25]

K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index,, J. Differential Equations, 221 (2006), 246. doi: 10.1016/j.jde.2005.03.006. Google Scholar

[26]

S. I. Pohožaev, On a class of quasilinear hyperbolic equations,, Mat. Sb. (N.S.) (Russian), 96 (1975), 152. Google Scholar

[27]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, CBMS Reg. Conf. Ser. Math., (1986). doi: 10.1090/cbms/065. Google Scholar

[28]

J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations,, Nonlinear Anal., 74 (2011), 1212. doi: 10.1016/j.na.2010.09.061. Google Scholar

[29]

M. Willem, Minimax Theorems,, Birkhäuser, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[30]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$,, Nonlinear Anal. Real World Appl., 12 (2011), 1278. doi: 10.1016/j.nonrwa.2010.09.023. Google Scholar

[31]

Y. Yang and J. H. Zhang, Nontrivial solutions of a class of nonlocal problems via local linking theory,, Appl. Math. Lett., 23 (2010), 377. doi: 10.1016/j.aml.2009.11.001. Google Scholar

[32]

Z. T. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow,, J. Math. Anal. Appl., 317 (2006), 456. doi: 10.1016/j.jmaa.2005.06.102. Google Scholar

show all references

References:
[1]

S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities,, Calc. Var. Partial Differential Equations, 1 (1993), 439. doi: 10.1007/BF01206962. Google Scholar

[2]

S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign,, J. Funct. Anal., 141 (1996), 159. doi: 10.1006/jfan.1996.0125. Google Scholar

[3]

C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,, Comput. Math. Appl., 49 (2005), 85. doi: 10.1016/j.camwa.2005.01.008. Google Scholar

[4]

C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbbR^N $,, Nonlinear Anal., 75 (2012), 2750. doi: 10.1016/j.na.2011.11.017. Google Scholar

[5]

P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data,, Invent. Math., 108 (1992), 247. doi: 10.1007/BF02100605. Google Scholar

[6]

A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string,, Trans. Amer. Math. Soc., 348 (1996), 305. doi: 10.1090/S0002-9947-96-01532-2. Google Scholar

[7]

A. Azzollini and P. d'Avenia, A. Pomponio, Multiple critical points for a class of nonlinear functionals,, Ann. Mat. Pura Appl., 190 (2011), 507. doi: 10.1007/s10231-010-0160-3. Google Scholar

[8]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions,, Arch. Rational Mech. Anal., 82 (1983), 347. doi: 10.1007/BF00250556. Google Scholar

[9]

S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées,, Bull. Acad. Sci. URSS, 4 (1940), 17. Google Scholar

[10]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation,, Adv. Differential Equations, 6 (2001), 701. Google Scholar

[11]

J. Q. Chen, Multiple positive solutions to a class of Kirchhoff equation on $\mathbbR^3$ with indefinite nonlinearity,, Nonlinear Analysis, 96 (2014), 134. doi: 10.1016/j.na.2013.11.012. Google Scholar

[12]

B. T. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems,, J. Math. Anal. Appl., 394 (2012), 488. doi: 10.1016/j.jmaa.2012.04.025. Google Scholar

[13]

S. Cingolani and J. L. Gomez, Positive solutions of a semilinear elliptic equation on $\mathbbR^N$ with indefinite nonlinearity,, Adv. Diff. Eq., 1 (1996), 773. Google Scholar

[14]

D. G. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in $\mathbbR^N$,, Calc. Var. Partial Differential Equations, 13 (2001), 159. doi: 10.1007/PL00009927. Google Scholar

[15]

G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities,, Archive for Rational Mechanics and Analysis, 213 (2014), 931. doi: 10.1007/s00205-014-0747-8. Google Scholar

[16]

Y. X. Guo and J. J. Nie, Existence and multiplicity of nontrivial solutions for $p$-Laplacian Schrödinger-Kirchhoff-type equations,, J. Math. Anal. Appl., 428 (2015), 1054. doi: 10.1016/j.jmaa.2015.03.064. Google Scholar

[17]

X. M. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems,, Nonlinear Anal., 70 (2009), 1407. doi: 10.1016/j.na.2008.02.021. Google Scholar

[18]

X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^3$,, J. Differential Equations, 252 (2012), 1813. doi: 10.1016/j.jde.2011.08.035. Google Scholar

[19]

J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$,, J. Math. Anal. Appl., 369 (2010), 564. doi: 10.1016/j.jmaa.2010.03.059. Google Scholar

[20]

G. Kirchhoff, Mechanik,, Teubner, (1883). Google Scholar

[21]

L. Liu and C. S. Chen, Study on existence of solutions for $p$-Kirchhoff elliptic equation in $\mathbbR^N$ with vanishing potential,, Journal of Dynamical and Control Systems, 20 (2014), 575. doi: 10.1007/s10883-014-9244-5. Google Scholar

[22]

W. Liu and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations,, J. Appl. Math. Comput., 39 (2012), 473. doi: 10.1007/s12190-012-0536-1. Google Scholar

[23]

J. L. Lions, On some questions in boundary value problems of mathematical physics,, North-Holland Math. Stud., 30 (1978), 284. Google Scholar

[24]

P. L. Lions, The Concentration-Compactness Principle in the Calculus of Variations. The Locally Compact Case, Part I,, Ann. Inst. H. Poincaré, 1 (1984), 109. Google Scholar

[25]

K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index,, J. Differential Equations, 221 (2006), 246. doi: 10.1016/j.jde.2005.03.006. Google Scholar

[26]

S. I. Pohožaev, On a class of quasilinear hyperbolic equations,, Mat. Sb. (N.S.) (Russian), 96 (1975), 152. Google Scholar

[27]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, CBMS Reg. Conf. Ser. Math., (1986). doi: 10.1090/cbms/065. Google Scholar

[28]

J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations,, Nonlinear Anal., 74 (2011), 1212. doi: 10.1016/j.na.2010.09.061. Google Scholar

[29]

M. Willem, Minimax Theorems,, Birkhäuser, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[30]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$,, Nonlinear Anal. Real World Appl., 12 (2011), 1278. doi: 10.1016/j.nonrwa.2010.09.023. Google Scholar

[31]

Y. Yang and J. H. Zhang, Nontrivial solutions of a class of nonlocal problems via local linking theory,, Appl. Math. Lett., 23 (2010), 377. doi: 10.1016/j.aml.2009.11.001. Google Scholar

[32]

Z. T. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow,, J. Math. Anal. Appl., 317 (2006), 456. doi: 10.1016/j.jmaa.2005.06.102. Google Scholar

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