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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 |
$ - (a + b\int_{{\mathbb{R}^3}} {|\nabla u{|^2}dx} )\Delta u + \lambda V(x)u = f(u),\;x \in {\mathbb{R}^3},$ |
$λ$ |
$V(x)$ |
$Ω: = int(V^{-1}(0))$ |
$k$ |
$Ω_1,Ω_2,···,Ω_k$ |
$f(u)$ |
$λ→ +∞$ |
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. 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. |
[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. 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. |
[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.
|
[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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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.
|
[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] | |
[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. 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. |
[36] | |
[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. 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. |
[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. 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. |
[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.
|
[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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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.
|
[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] | |
[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. 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. |
[36] | |
[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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