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:
[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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

G. Kirchhoff, Mechanik Teubner, Leipzig, 1883.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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.

[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.

[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.

[36]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

G. Kirchhoff, Mechanik Teubner, Leipzig, 1883.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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.

[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.

[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.

[36]

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

[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.

[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.

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