August  2022, 21(8): 2495-2528. doi: 10.3934/cpaa.2022058

Multiple localized nodal solutions of high topological type for Kirchhoff-type equation with double potentials

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, China

*Corresponding author

Received  October 2021 Revised  January 2022 Published  August 2022 Early access  March 2022

Fund Project: This work is supported by National Natural Science Foundation of China (No. 11961043 and 11561043)

We are concerned with sign-changing solutions and their concentration behaviors of singularly perturbed Kirchhoff problem
$ \begin{equation*} -(\varepsilon^{2}a+ \varepsilon b\int _{\mathbb{R}^{3}}|\nabla v|^{2}dx)\Delta v+V(x)v = P(x)f(v)\; \; {\rm{in}}\; \mathbb{R}^{3}, \end{equation*} $
where
$ \varepsilon $
is a small positive parameter,
$ a, b>0 $
and
$ V, P\in C^{1}(\mathbb{R}^{3}, \mathbb{R}) $
. Without using any non-degeneracy conditions, we obtain multiple localized sign-changing solutions of higher topological type for this problem. Furthermore, we also determine a concrete set as the concentration position of these sign-changing solutions. The main methods we use are penalization techniques and the method of invariant sets of descending flow. It is notice that, when nonlinear potential
$ P $
is a positive constant, our result generalizes the result obtained in [5] to Kirchhoff problem.
Citation: Zhi-Guo Wu, Wen Guan, Da-Bin Wang. Multiple localized nodal solutions of high topological type for Kirchhoff-type equation with double potentials. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2495-2528. doi: 10.3934/cpaa.2022058
References:
[1]

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

[2]

J. Byeon and Z. Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Ⅱ, Calc. Var. Partial Differ. Equ., 18 (2003), 207-219.  doi: 10.1007/s00526-002-0191-8.

[3]

D. CassaniZ. LiuC. Tarsi and J. J. Zhang, Multiplicity of sign-changing solutions for Kirchhoff-type equations, Nonlinear Anal., 186 (2019), 145-161.  doi: 10.1016/j.na.2019.01.025.

[4]

S. ChenJ. Q. Liu and Z. Q. Wang, Localized nodal solutions for a critical nonlinear Schrödinger equation, J. Funct. Anal., 277 (2019), 594-640.  doi: 10.1016/j.jfa.2018.10.027.

[5]

S. Chen and Z. Q. Wang, Localized nodal solutions of higher topological type for semiclassical nonlinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 56 (2017), 1-26.  doi: 10.1007/s00526-016-1094-4.

[6]

T. D'Aprile and A. Pistoia, Existence multiplicity and profile of sign-changing clustered solutions of a semiclassical nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1423-1451.  doi: 10.1016/j.anihpc.2009.01.002.

[7]

Y. B. DengS. J. 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.

[8]

A. Floer and A. Weinstein, Nonspreading wave pachets for the packets for the cubic Schrödinger with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.

[9]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[10]

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

[11]

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

[12]

X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differ. Equ., 5 (2000), 899-928. 

[13]

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

[14]

G. B. LiP. LuoS. J. PengC. H. Wang and C. L. Xiang, A singularly perturbed Kirchhoff problem revisited, J. Differ. Equ., 268 (2020), 541-589.  doi: 10.1016/j.jde.2019.08.016.

[15]

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

[16]

Y. H. LiF. Y. Li and J. P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equ., 253 (2012), 2285-2294.  doi: 10.1016/j.jde.2012.05.017.

[17]

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

[18]

J. Q. LiuX. Liu and Z. Q. Wang, Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var. Partial Differ. Equ., 52 (2015), 565-586.  doi: 10.1007/s00526-014-0724-y.

[19]

J. Q. LiuX. Liu and Z. Q. Wang, Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth, J. Differ. Equ., 261 (2016), 7194-7236.  doi: 10.1016/j.jde.2016.09.018.

[20]

X. LiuJ. Q. Liu and Z. Q. Wang, Localized nodal solutions for quasilinear Schrödinger equations, J. Differ. Equ., 267 (2019), 7411-7461.  doi: 10.1016/j.jde.2019.08.003.

[21]

Z. Liu, Y. Lou and J. J. Zhang, A perturbation approach to studying sign-changing solutions of Kirchhoff equations with a general nonlinearity, arXiv: 1812.09240v2.

[22]

D. Oplinger, Frequency response of a nonlinear stretched string, J. Acoust. Soc. Amer., 32 (1960), 1529-1538.  doi: 10.1121/1.1907948.

[23]

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

[24]

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

[25]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.

[26]

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

[27]

J. SunL. LiM. Cencelj and B. Gabrovšek, Infinitely many sign-changing solutions for Kirchhoff type problems in $\mathbb{R}^{3}$, Nonlinear Anal., 186 (2019), 33-54.  doi: 10.1016/j.na.2018.10.007.

[28]

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

[29] K. Tintarev and K. H. Fieseler, Concentration Compactness Functional-Analytic Grounds and Applications, Imperial College Press, London, 2007.  doi: 10.1142/p456.
[30]

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

[31]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244. 

[32]

L. WangB. L. Zhang and K. Cheng, Ground state sign-changing solutions for the Schrödinger-Kirchhoff equation in $\mathbb{R}^{3}$, J. Math. Anal. Appl., 466 (2018), 1545-1569.  doi: 10.1016/j.jmaa.2018.06.071.

[33]

Q. L. XieS. W. Ma and X. Zhang, Bound state solutions of Kirchhoff type problems with critical exponent, J. Differ. Equ., 261 (2016), 890-924.  doi: 10.1016/j.jde.2016.03.028.

[34]

Q. L. Xie and X. Zhang, Semi-classical solutions for Kirchhoff type problem with a critical frequency, Proc. Roy. Soc. Edinb., 151 (2021), 761-798.  doi: 10.1017/prm.2020.37.

[35]

Y. Yu and Y. H. Ding, An infinite sequence of localized nodal solutions for Schrödinger-Poisson system with double potentials, arXiv: 2007.14599v1.

show all references

References:
[1]

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

[2]

J. Byeon and Z. Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Ⅱ, Calc. Var. Partial Differ. Equ., 18 (2003), 207-219.  doi: 10.1007/s00526-002-0191-8.

[3]

D. CassaniZ. LiuC. Tarsi and J. J. Zhang, Multiplicity of sign-changing solutions for Kirchhoff-type equations, Nonlinear Anal., 186 (2019), 145-161.  doi: 10.1016/j.na.2019.01.025.

[4]

S. ChenJ. Q. Liu and Z. Q. Wang, Localized nodal solutions for a critical nonlinear Schrödinger equation, J. Funct. Anal., 277 (2019), 594-640.  doi: 10.1016/j.jfa.2018.10.027.

[5]

S. Chen and Z. Q. Wang, Localized nodal solutions of higher topological type for semiclassical nonlinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 56 (2017), 1-26.  doi: 10.1007/s00526-016-1094-4.

[6]

T. D'Aprile and A. Pistoia, Existence multiplicity and profile of sign-changing clustered solutions of a semiclassical nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1423-1451.  doi: 10.1016/j.anihpc.2009.01.002.

[7]

Y. B. DengS. J. 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.

[8]

A. Floer and A. Weinstein, Nonspreading wave pachets for the packets for the cubic Schrödinger with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.

[9]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[10]

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

[11]

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

[12]

X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differ. Equ., 5 (2000), 899-928. 

[13]

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

[14]

G. B. LiP. LuoS. J. PengC. H. Wang and C. L. Xiang, A singularly perturbed Kirchhoff problem revisited, J. Differ. Equ., 268 (2020), 541-589.  doi: 10.1016/j.jde.2019.08.016.

[15]

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

[16]

Y. H. LiF. Y. Li and J. P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equ., 253 (2012), 2285-2294.  doi: 10.1016/j.jde.2012.05.017.

[17]

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

[18]

J. Q. LiuX. Liu and Z. Q. Wang, Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var. Partial Differ. Equ., 52 (2015), 565-586.  doi: 10.1007/s00526-014-0724-y.

[19]

J. Q. LiuX. Liu and Z. Q. Wang, Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth, J. Differ. Equ., 261 (2016), 7194-7236.  doi: 10.1016/j.jde.2016.09.018.

[20]

X. LiuJ. Q. Liu and Z. Q. Wang, Localized nodal solutions for quasilinear Schrödinger equations, J. Differ. Equ., 267 (2019), 7411-7461.  doi: 10.1016/j.jde.2019.08.003.

[21]

Z. Liu, Y. Lou and J. J. Zhang, A perturbation approach to studying sign-changing solutions of Kirchhoff equations with a general nonlinearity, arXiv: 1812.09240v2.

[22]

D. Oplinger, Frequency response of a nonlinear stretched string, J. Acoust. Soc. Amer., 32 (1960), 1529-1538.  doi: 10.1121/1.1907948.

[23]

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

[24]

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

[25]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.

[26]

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

[27]

J. SunL. LiM. Cencelj and B. Gabrovšek, Infinitely many sign-changing solutions for Kirchhoff type problems in $\mathbb{R}^{3}$, Nonlinear Anal., 186 (2019), 33-54.  doi: 10.1016/j.na.2018.10.007.

[28]

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

[29] K. Tintarev and K. H. Fieseler, Concentration Compactness Functional-Analytic Grounds and Applications, Imperial College Press, London, 2007.  doi: 10.1142/p456.
[30]

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

[31]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244. 

[32]

L. WangB. L. Zhang and K. Cheng, Ground state sign-changing solutions for the Schrödinger-Kirchhoff equation in $\mathbb{R}^{3}$, J. Math. Anal. Appl., 466 (2018), 1545-1569.  doi: 10.1016/j.jmaa.2018.06.071.

[33]

Q. L. XieS. W. Ma and X. Zhang, Bound state solutions of Kirchhoff type problems with critical exponent, J. Differ. Equ., 261 (2016), 890-924.  doi: 10.1016/j.jde.2016.03.028.

[34]

Q. L. Xie and X. Zhang, Semi-classical solutions for Kirchhoff type problem with a critical frequency, Proc. Roy. Soc. Edinb., 151 (2021), 761-798.  doi: 10.1017/prm.2020.37.

[35]

Y. Yu and Y. H. Ding, An infinite sequence of localized nodal solutions for Schrödinger-Poisson system with double potentials, arXiv: 2007.14599v1.

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