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 and Continuous Dynamical Systems - S, 2016, 9 (6) : 1663-1685. doi: 10.3934/dcdss.2016069
References:
[1]

S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475. doi: 10.1007/BF01206962.

[2]

S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215. doi: 10.1006/jfan.1996.0125.

[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), 85-93. doi: 10.1016/j.camwa.2005.01.008.

[4]

C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbbR^N $, Nonlinear Anal., 75 (2012), 2750-2759. doi: 10.1016/j.na.2011.11.017.

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P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262. doi: 10.1007/BF02100605.

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A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330. doi: 10.1090/S0002-9947-96-01532-2.

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A. Azzollini and P. d'Avenia, A. Pomponio, Multiple critical points for a class of nonlinear functionals, Ann. Mat. Pura Appl., 190 (2011), 507-523. doi: 10.1007/s10231-010-0160-3.

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H. Berestycki and P. L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556.

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S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées, Bull. Acad. Sci. URSS, Ser. Mat., 4 (1940), 17-26 (Izvestia Akad. Nauk SSSR).

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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), 701-730.

[11]

J. Q. Chen, Multiple positive solutions to a class of Kirchhoff equation on $\mathbbR^3$ with indefinite nonlinearity, Nonlinear Analysis, 96 (2014), 134-145. doi: 10.1016/j.na.2013.11.012.

[12]

B. T. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl., 394 (2012), 488-495. doi: 10.1016/j.jmaa.2012.04.025.

[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), 773-791.

[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), 159-189. doi: 10.1007/PL00009927.

[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), 931-979. doi: 10.1007/s00205-014-0747-8.

[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), 1054-1069. doi: 10.1016/j.jmaa.2015.03.064.

[17]

X. M. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414. doi: 10.1016/j.na.2008.02.021.

[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), 1813-1834. doi: 10.1016/j.jde.2011.08.035.

[19]

J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$, J. Math. Anal. Appl., 369 (2010), 564-574. doi: 10.1016/j.jmaa.2010.03.059.

[20]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.

[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), 575-592. doi: 10.1007/s10883-014-9244-5.

[22]

W. Liu and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487. doi: 10.1007/s12190-012-0536-1.

[23]

J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346.

[24]

P. L. Lions, The Concentration-Compactness Principle in the Calculus of Variations. The Locally Compact Case, Part I, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 1 (1984), 109-145.

[25]

K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221 (2006), 246-255. doi: 10.1016/j.jde.2005.03.006.

[26]

S. I. Pohožaev, On a class of quasilinear hyperbolic equations, Mat. Sb. (N.S.) (Russian), 96 (1975), 152-166(in Russian).

[27]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986. doi: 10.1090/cbms/065.

[28]

J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222. doi: 10.1016/j.na.2010.09.061.

[29]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[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), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.

[31]

Y. Yang and J. H. Zhang, Nontrivial solutions of a class of nonlocal problems via local linking theory, Appl. Math. Lett., 23 (2010), 377-380. doi: 10.1016/j.aml.2009.11.001.

[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), 456-463. doi: 10.1016/j.jmaa.2005.06.102.

show all references

References:
[1]

S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475. doi: 10.1007/BF01206962.

[2]

S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215. doi: 10.1006/jfan.1996.0125.

[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), 85-93. doi: 10.1016/j.camwa.2005.01.008.

[4]

C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbbR^N $, Nonlinear Anal., 75 (2012), 2750-2759. doi: 10.1016/j.na.2011.11.017.

[5]

P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262. doi: 10.1007/BF02100605.

[6]

A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330. doi: 10.1090/S0002-9947-96-01532-2.

[7]

A. Azzollini and P. d'Avenia, A. Pomponio, Multiple critical points for a class of nonlinear functionals, Ann. Mat. Pura Appl., 190 (2011), 507-523. doi: 10.1007/s10231-010-0160-3.

[8]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556.

[9]

S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées, Bull. Acad. Sci. URSS, Ser. Mat., 4 (1940), 17-26 (Izvestia Akad. Nauk SSSR).

[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), 701-730.

[11]

J. Q. Chen, Multiple positive solutions to a class of Kirchhoff equation on $\mathbbR^3$ with indefinite nonlinearity, Nonlinear Analysis, 96 (2014), 134-145. doi: 10.1016/j.na.2013.11.012.

[12]

B. T. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl., 394 (2012), 488-495. doi: 10.1016/j.jmaa.2012.04.025.

[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), 773-791.

[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), 159-189. doi: 10.1007/PL00009927.

[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), 931-979. doi: 10.1007/s00205-014-0747-8.

[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), 1054-1069. doi: 10.1016/j.jmaa.2015.03.064.

[17]

X. M. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414. doi: 10.1016/j.na.2008.02.021.

[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), 1813-1834. doi: 10.1016/j.jde.2011.08.035.

[19]

J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$, J. Math. Anal. Appl., 369 (2010), 564-574. doi: 10.1016/j.jmaa.2010.03.059.

[20]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.

[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), 575-592. doi: 10.1007/s10883-014-9244-5.

[22]

W. Liu and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487. doi: 10.1007/s12190-012-0536-1.

[23]

J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346.

[24]

P. L. Lions, The Concentration-Compactness Principle in the Calculus of Variations. The Locally Compact Case, Part I, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 1 (1984), 109-145.

[25]

K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221 (2006), 246-255. doi: 10.1016/j.jde.2005.03.006.

[26]

S. I. Pohožaev, On a class of quasilinear hyperbolic equations, Mat. Sb. (N.S.) (Russian), 96 (1975), 152-166(in Russian).

[27]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986. doi: 10.1090/cbms/065.

[28]

J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222. doi: 10.1016/j.na.2010.09.061.

[29]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[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), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.

[31]

Y. Yang and J. H. Zhang, Nontrivial solutions of a class of nonlocal problems via local linking theory, Appl. Math. Lett., 23 (2010), 377-380. doi: 10.1016/j.aml.2009.11.001.

[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), 456-463. doi: 10.1016/j.jmaa.2005.06.102.

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