June  2018, 38(6): 3139-3168. doi: 10.3934/dcds.2018137

Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$

1. 

School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China

2. 

The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China

* Corresponding author: Wei Shuai

Received  November 2017 Revised  December 2017 Published  April 2018

We are interested in the existence of sign-changing multi-bump solutions for the following Kirchhoff equation
$ - (a + b\int_{{\mathbb{R}^3}} {|\nabla u{|^2}dx} )\Delta u + \lambda V(x)u = f(u),\;x \in {\mathbb{R}^3},$
where
$λ$
>0 is a parameter and the potential
$V(x)$
is a nonnegative continuous function with a potential well
$Ω: = int(V^{-1}(0))$
which possesses
$k$
disjoint bounded components
$Ω_1,Ω_2,···,Ω_k$
. Under some conditions imposed on
$f(u)$
, multiple sign-changing multi-bump solutions are obtained. Moreover, the concentration behavior of these solutions as
$λ→ +∞$
are also studied.
Citation: Yinbin Deng, Wei Shuai. Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3139-3168. doi: 10.3934/dcds.2018137
References:
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[2]

C. Alves, Multiplicity of multi-bump type nodal solutions for a class of elliptic problems in $\mathbb{R}^N$, Topol. Methods Nonlinear Anal., 34 (2009), 231-250. doi: 10.12775/TMNA.2009.040. Google Scholar

[3]

C. Alves and G. Figueiredo, Multi-bump solutions for a Kirchhoff-type problem, Adv. Nonlinear Anal., 5 (2016), 1-26. doi: 10.1515/anona-2015-0101. Google Scholar

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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. Google Scholar

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T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149. Google Scholar

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T. BartschA. Pankov and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494. Google Scholar

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H. Berestycki and P. L. Lions, Nonlinear scalar field equations Ⅰ, Arch. Ration. Mech. Anal., 82 (1983), 313-345. Google Scholar

[8]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations Ⅱ, Arch. Ration. Mech. Anal., 82 (1983), 347-375. Google Scholar

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A. CastroJ. Cossio and J. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math., 27 (1997), 1041-1053. doi: 10.1216/rmjm/1181071858. Google Scholar

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M. CavalcantiV. Domingos Cavalcanti and J. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6 (2001), 701-730. Google Scholar

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C. ChenY. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017. Google Scholar

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M. ContiS. Terracini and G. Verzini, Nehari's problem and competing species systems, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire., 19 (2002), 871-888. doi: 10.1016/S0294-1449(02)00104-X. Google Scholar

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M. del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137. doi: 10.1007/BF01189950. Google Scholar

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Y. DengS. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}^3$, J. Funct. Anal., 269 (2015), 3500-3527. doi: 10.1016/j.jfa.2015.09.012. Google Scholar

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Y. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manuscripta Math., 112 (2003), 109-135. doi: 10.1007/s00229-003-0397-x. Google Scholar

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G. FigueiredoN. Ikoma and J. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8. Google Scholar

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G. Figueiredo and R. Nascimento, Existence of a nodal solution with minimal energy for a Kirchhoff equation, Math. Nachr., 288 (2015), 48-60. doi: 10.1002/mana.201300195. Google Scholar

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G. Figueiredo and J. Santos, Existence of a least energy nodal solution for a Schrödinger-Kirchhoff equation with potential vanishing at infinity, J. Math. Phys. 56 (2015), 051506, 18 pp. Google Scholar

[21]

X. He and W. Zou, Existence and concentration behavior of positive solutions for a kirchhoff equation in $\mathbb{R}^3$, J. Differential Equations, 252 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035. Google Scholar

[22]

Y. HeG. Li and S. Peng, Concentrating bound states for Kirchhoff type problems in $\mathbb{R}^3$ involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (2014), 483-510. Google Scholar

[23]

Y. He and G. Li, Standing waves for a class of Kirchhoff type problems in $\mathbb{R}^3$ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106. doi: 10.1007/s00526-015-0894-2. Google Scholar

[24]

Y. He, Concentrating bounded states for a class of singularly perturbed Kirchhoff type equations with a general nonlinearity, J. Differential Equations, 261 (2016), 6178-6220. doi: 10.1016/j.jde.2016.08.034. Google Scholar

[25]

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

[26]

G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^3$, J. Differential Equations, 257 (2014), 566-600. doi: 10.1016/j.jde.2014.04.011. Google Scholar

[27]

J. L. Lions, On some questions in boundary value problems of mathematical physics, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, New York, 30, (1978), 284-346. Google Scholar

[28]

S. Lu, Signed and sign-changing solutions for a Kirchhoff-type equation in bounded domains, J. Math. Anal. Appl., 432 (2015), 965-982. doi: 10.1016/j.jmaa.2015.07.033. Google Scholar

[29]

K. Perera and Z. 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. Google Scholar

[30]

Y. Sato and K. Tanaka, Sign-changing multi-bump solutions for nonlinear Schrödinger equations with steep potential wells, Trans. Amer. Math. Soc., 361 (2009), 6205-6253. doi: 10.1090/S0002-9947-09-04565-6. Google Scholar

[31]

W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equations, 259 (2015), 1256-1274. doi: 10.1016/j.jde.2015.02.040. Google Scholar

[32]

W. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517. Google Scholar

[33]

J. Sun and T. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations, 256 (2014), 1771-1792. doi: 10.1016/j.jde.2013.12.006. Google Scholar

[34]

X. Tang and B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402. doi: 10.1016/j.jde.2016.04.032. Google Scholar

[35]

J. WangL. TianJ. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351. doi: 10.1016/j.jde.2012.05.023. Google Scholar

[36]

M. Willem, Minimax Theorems Birkhäuser, Barel, 1996. Google Scholar

[37]

H. Ye, The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations in $\mathbb{R}^N$, J. Math. Anal. Appl., 431 (2015), 935-954. doi: 10.1016/j.jmaa.2015.06.012. Google Scholar

[38]

Z. 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. Google Scholar

show all references

References:
[1]

C. Alves and F. Correa, On existence of solutions for a class of problem involving a nonlinear operator, Comm. Appl. Nonlinear Anal., 8 (2001), 43-56. Google Scholar

[2]

C. Alves, Multiplicity of multi-bump type nodal solutions for a class of elliptic problems in $\mathbb{R}^N$, Topol. Methods Nonlinear Anal., 34 (2009), 231-250. doi: 10.12775/TMNA.2009.040. Google Scholar

[3]

C. Alves and G. Figueiredo, Multi-bump solutions for a Kirchhoff-type problem, Adv. Nonlinear Anal., 5 (2016), 1-26. doi: 10.1515/anona-2015-0101. Google Scholar

[4]

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. Google Scholar

[5]

T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149. Google Scholar

[6]

T. BartschA. Pankov and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494. Google Scholar

[7]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations Ⅰ, Arch. Ration. Mech. Anal., 82 (1983), 313-345. Google Scholar

[8]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations Ⅱ, Arch. Ration. Mech. Anal., 82 (1983), 347-375. Google Scholar

[9]

A. CastroJ. Cossio and J. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math., 27 (1997), 1041-1053. doi: 10.1216/rmjm/1181071858. Google Scholar

[10]

M. CavalcantiV. Domingos Cavalcanti and J. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6 (2001), 701-730. Google Scholar

[11]

C. ChenY. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017. Google Scholar

[12]

M. ContiS. Terracini and G. Verzini, Nehari's problem and competing species systems, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire., 19 (2002), 871-888. doi: 10.1016/S0294-1449(02)00104-X. Google Scholar

[13]

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. Google Scholar

[14]

M. del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137. doi: 10.1007/BF01189950. Google Scholar

[15]

Y. DengS. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}^3$, J. Funct. Anal., 269 (2015), 3500-3527. doi: 10.1016/j.jfa.2015.09.012. Google Scholar

[16]

Y. DengS. Peng and W. Shuai, Nodal standing waves for a gauged nonlinear Schrödinger equation in $\mathbb{R}^2$, J. Differential Equations, 264 (2018), 4006-4035. doi: 10.1016/j.jde.2017.12.003. Google Scholar

[17]

Y. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manuscripta Math., 112 (2003), 109-135. doi: 10.1007/s00229-003-0397-x. Google Scholar

[18]

G. FigueiredoN. Ikoma and J. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8. Google Scholar

[19]

G. Figueiredo and R. Nascimento, Existence of a nodal solution with minimal energy for a Kirchhoff equation, Math. Nachr., 288 (2015), 48-60. doi: 10.1002/mana.201300195. Google Scholar

[20]

G. Figueiredo and J. Santos, Existence of a least energy nodal solution for a Schrödinger-Kirchhoff equation with potential vanishing at infinity, J. Math. Phys. 56 (2015), 051506, 18 pp. Google Scholar

[21]

X. He and W. Zou, Existence and concentration behavior of positive solutions for a kirchhoff equation in $\mathbb{R}^3$, J. Differential Equations, 252 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035. Google Scholar

[22]

Y. HeG. Li and S. Peng, Concentrating bound states for Kirchhoff type problems in $\mathbb{R}^3$ involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (2014), 483-510. Google Scholar

[23]

Y. He and G. Li, Standing waves for a class of Kirchhoff type problems in $\mathbb{R}^3$ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106. doi: 10.1007/s00526-015-0894-2. Google Scholar

[24]

Y. He, Concentrating bounded states for a class of singularly perturbed Kirchhoff type equations with a general nonlinearity, J. Differential Equations, 261 (2016), 6178-6220. doi: 10.1016/j.jde.2016.08.034. Google Scholar

[25]

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

[26]

G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^3$, J. Differential Equations, 257 (2014), 566-600. doi: 10.1016/j.jde.2014.04.011. Google Scholar

[27]

J. L. Lions, On some questions in boundary value problems of mathematical physics, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, New York, 30, (1978), 284-346. Google Scholar

[28]

S. Lu, Signed and sign-changing solutions for a Kirchhoff-type equation in bounded domains, J. Math. Anal. Appl., 432 (2015), 965-982. doi: 10.1016/j.jmaa.2015.07.033. Google Scholar

[29]

K. Perera and Z. 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. Google Scholar

[30]

Y. Sato and K. Tanaka, Sign-changing multi-bump solutions for nonlinear Schrödinger equations with steep potential wells, Trans. Amer. Math. Soc., 361 (2009), 6205-6253. doi: 10.1090/S0002-9947-09-04565-6. Google Scholar

[31]

W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equations, 259 (2015), 1256-1274. doi: 10.1016/j.jde.2015.02.040. Google Scholar

[32]

W. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517. Google Scholar

[33]

J. Sun and T. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations, 256 (2014), 1771-1792. doi: 10.1016/j.jde.2013.12.006. Google Scholar

[34]

X. Tang and B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402. doi: 10.1016/j.jde.2016.04.032. Google Scholar

[35]

J. WangL. TianJ. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351. doi: 10.1016/j.jde.2012.05.023. Google Scholar

[36]

M. Willem, Minimax Theorems Birkhäuser, Barel, 1996. Google Scholar

[37]

H. Ye, The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations in $\mathbb{R}^N$, J. Math. Anal. Appl., 431 (2015), 935-954. doi: 10.1016/j.jmaa.2015.06.012. Google Scholar

[38]

Z. 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. Google Scholar

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