# American Institute of Mathematical Sciences

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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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. Castro, J. 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. Cavalcanti, V. 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. Chen, Y. 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. Conti, S. 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. Deng, S. 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. Deng, S. 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. Figueiredo, N. 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. He, G. 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. Wang, L. Tian, J. 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. Bartsch, A. 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. Castro, J. 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. Cavalcanti, V. 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. Chen, Y. 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. Conti, S. 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. Deng, S. 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. Deng, S. 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. Figueiredo, N. 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. He, G. 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. Wang, L. Tian, J. 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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