The effect of the weight function on the number of nodal solutions of the Kirchhoff-type equations in high dimensions

Previous Article
Stability, existence and non-existence of $ T $-periodic solutions of nonlinear delayed differential equations with $ \varphi $-Laplacian
CPAA Home
This Issue
Next Article
On the solvability of a semilinear higher-order elliptic problem for the vector field method in image registration

August 2022, 21(8): 2701-2721. doi: 10.3934/cpaa.2022069

The effect of the weight function on the number of nodal solutions of the Kirchhoff-type equations in high dimensions

He Zhang and Haibo Chen ^,

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

^* Corresponding author

Received July 2021 Revised February 2022 Published August 2022 Early access April 2022

Fund Project: H. Chen is supported by the National Natural Science Foundation of China (Grant No. 12071486)

In this paper, we consider the multiplicity of nodal solutions for the following Kirchhoff type equations:

$ \begin{equation*} \left\{ \begin{array}{ll} -\varepsilon^2M\left(\varepsilon^{2-N}||\nabla u||^2_{L^2}\right)\Delta u+u = f\left(x\right)|u|^{p-2}u,\ \text{in}\ \mathbb{R}^N,\\ u\in H^1(\mathbb{R}^N), \end{array} \right. \end{equation*} $

where

$ N\geq 4 $

,

$ \varepsilon>0 $

is a small parameter,

$ M\left(t\right) = at+b\left(a,b>0\right) $

and

$ 2<p<2^* = \frac{2N}{N-2} $

. We assume that the weight function

$ f\in C\left(\mathbb{R}^N,\mathbb{R}^+\right) $

has

$ k $

maximum points in

$ \mathbb{R}^N $

. By using a novel constraint approach as well as the barycenter map,

$ k^2 $

nodal solutions are obtained when

$ N\geq4 $

for

$ \varepsilon,a $

sufficiently small.

Keywords: Kirchhoff–type problem, nodal solutions, variational method, the barycenter map.

Mathematics Subject Classification: 35J20; 35J61.

Citation: He Zhang, Haibo Chen. The effect of the weight function on the number of nodal solutions of the Kirchhoff-type equations in high dimensions. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2701-2721. doi: 10.3934/cpaa.2022069

References:

[1]	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.
[2]	T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281. doi: 10.1016/j.anihpc.2004.07.005.
[3]	S. Bernstein, Sur une class d'équations fonctionnelles aux dérivés partielles, Bull. Acad. Sci. URSS, Sr. Math., 4 (1940), 17-26.
[4]	V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. doi: 10.12775/TMNA.1998.019.
[5]	H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.
[6]	G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differ. Equ., 17 (2003), 257-281. doi: 10.1007/s00526-002-0169-6.
[7]	M. Clapp and T. Weth, Minimal nodal solutions of the pure critical exponent problem on a symmetric doamin, Calc. Var. Partial Differ. Equ., 21 (2004), 1-14. doi: 10.1007/s00526-003-0241-x.
[8]	Y. B. Deng, S. 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.
[9]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.
[10]	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.
[11]	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, 18pp. doi: 10.1063/1.4921639.
[12]	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. doi: 10.1515/ans-2014-0214.
[13]	T. Hu and L. Lu, Multiplicity of positive solutions for Kirchhoff type problems in $\mathbb{R}^3$, Topol. Methods Nonlinear Anal., 50 (2017), 231-252. doi: 10.12775/tmna.2017.028.
[14]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[15]	J. L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, 1978,284–346.
[16]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case II, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 1 (1984), 223-283.
[17]	Y. Li and W. M. Ni, Radial symmetry of positive solutions of nonlinear ellitic equations in $\mathbb{R}^N$, Commun. Partial Differ. Equ., 18 (1993), 1043-1054. doi: 10.1080/03605309308820960.
[18]	C. Liu, H. Wang and T. F. Wu, Multiplicity of 2-nodal solutions for semilinear elliptic problems in $\mathbb{R}^N$, J. Math. Anal. Appl., 348 (2008), 169-179. doi: 10.1016/j.jmaa.2008.06.042.
[19]	S. I. Pohozaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (NS), 96 (1975), 152-168.
[20]	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.
[21]	J. T. Sun, T. F. Wu and Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger-Poisson system, J. Differ. Equ., 260 (2016), 586-627. doi: 10.1016/j.jde.2015.09.002.
[22]	J. T. Sun and T. F. Wu, On the Kirchhoff type equations in $\mathbb{R}^N$, 2019, arXiv: 1908.01326.
[23]	J. T. Sun and T. F. Wu, Bound state nodal solutions for the non-autonomous Schrödinger–Poisson system in $\mathbb{R}^3$, J. Differ. Equ., 268 (2020), 7121-7163. doi: 10.1016/j.jde.2019.11.070.
[24]	J. T. Sun and T. F. Wu, The number of nodal solutions for the schrödinger-pisson system under the effect of the weight function, Discrete Contin. Dyn. Syst., 41 (2021), 3651-3682. doi: 10.3934/dcds.2021011.
[25]	G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 9 (1992), 281-304. doi: 10.1016/S0294-1449(16)30238-4.
[26]	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.
[27]	E. Zeidler, Nonlinear Functional Analysis and Its Applications I, Fixed–point Theorems, Springer, New York, 1986.
[28]	Z. T. 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.
[29]	Zhang, J., Sun, J., Wu, T. F., The number of positive solutions affected by the weight function to Kirchhoff type equations in high dimensions, Nonlinear Anal., 196 (2020), 111780, 24 pp. doi: 10.1016/j.na.2020.111780.

show all references

References:

[1]	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.
[2]	T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281. doi: 10.1016/j.anihpc.2004.07.005.
[3]	S. Bernstein, Sur une class d'équations fonctionnelles aux dérivés partielles, Bull. Acad. Sci. URSS, Sr. Math., 4 (1940), 17-26.
[4]	V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. doi: 10.12775/TMNA.1998.019.
[5]	H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.
[6]	G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differ. Equ., 17 (2003), 257-281. doi: 10.1007/s00526-002-0169-6.
[7]	M. Clapp and T. Weth, Minimal nodal solutions of the pure critical exponent problem on a symmetric doamin, Calc. Var. Partial Differ. Equ., 21 (2004), 1-14. doi: 10.1007/s00526-003-0241-x.
[8]	Y. B. Deng, S. 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.
[9]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.
[10]	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.
[11]	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, 18pp. doi: 10.1063/1.4921639.
[12]	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. doi: 10.1515/ans-2014-0214.
[13]	T. Hu and L. Lu, Multiplicity of positive solutions for Kirchhoff type problems in $\mathbb{R}^3$, Topol. Methods Nonlinear Anal., 50 (2017), 231-252. doi: 10.12775/tmna.2017.028.
[14]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[15]	J. L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, 1978,284–346.
[16]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case II, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 1 (1984), 223-283.
[17]	Y. Li and W. M. Ni, Radial symmetry of positive solutions of nonlinear ellitic equations in $\mathbb{R}^N$, Commun. Partial Differ. Equ., 18 (1993), 1043-1054. doi: 10.1080/03605309308820960.
[18]	C. Liu, H. Wang and T. F. Wu, Multiplicity of 2-nodal solutions for semilinear elliptic problems in $\mathbb{R}^N$, J. Math. Anal. Appl., 348 (2008), 169-179. doi: 10.1016/j.jmaa.2008.06.042.
[19]	S. I. Pohozaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (NS), 96 (1975), 152-168.
[20]	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.
[21]	J. T. Sun, T. F. Wu and Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger-Poisson system, J. Differ. Equ., 260 (2016), 586-627. doi: 10.1016/j.jde.2015.09.002.
[22]	J. T. Sun and T. F. Wu, On the Kirchhoff type equations in $\mathbb{R}^N$, 2019, arXiv: 1908.01326.
[23]	J. T. Sun and T. F. Wu, Bound state nodal solutions for the non-autonomous Schrödinger–Poisson system in $\mathbb{R}^3$, J. Differ. Equ., 268 (2020), 7121-7163. doi: 10.1016/j.jde.2019.11.070.
[24]	J. T. Sun and T. F. Wu, The number of nodal solutions for the schrödinger-pisson system under the effect of the weight function, Discrete Contin. Dyn. Syst., 41 (2021), 3651-3682. doi: 10.3934/dcds.2021011.
[25]	G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 9 (1992), 281-304. doi: 10.1016/S0294-1449(16)30238-4.
[26]	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.
[27]	E. Zeidler, Nonlinear Functional Analysis and Its Applications I, Fixed–point Theorems, Springer, New York, 1986.
[28]	Z. T. 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.
[29]	Zhang, J., Sun, J., Wu, T. F., The number of positive solutions affected by the weight function to Kirchhoff type equations in high dimensions, Nonlinear Anal., 196 (2020), 111780, 24 pp. doi: 10.1016/j.na.2020.111780.

[1]	Xiao-Jing Zhong, Chun-Lei Tang. The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem. Communications on Pure and Applied Analysis, 2017, 16 (2) : 611-628. doi: 10.3934/cpaa.2017030
[2]	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
[3]	Jun Wang, Lu Xiao. Existence and concentration of solutions for a Kirchhoff type problem with potentials. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7137-7168. doi: 10.3934/dcds.2016111
[4]	Wenjing Chen. Multiplicity of solutions for a fractional Kirchhoff type problem. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2009-2020. doi: 10.3934/cpaa.2015.14.2009
[5]	Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773
[6]	Rui-Qi Liu, Chun-Lei Tang, Jia-Feng Liao, Xing-Ping Wu. Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1841-1856. doi: 10.3934/cpaa.2016006
[7]	Yuanxiao Li, Ming Mei, Kaijun Zhang. Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 883-908. doi: 10.3934/dcdsb.2016.21.883
[8]	Alfonso Castro, Shu-Zhi Song. Infinitely many radial solutions for a super-cubic Kirchhoff type problem in a ball. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3347-3355. doi: 10.3934/dcdss.2020127
[9]	Jacob Ashiwere Abuchu, Godwin Chidi Ugwunnadi, Ojen Kumar Narain. Inertial Mann-Type iterative method for solving split monotone variational inclusion problem with applications. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022075
[10]	Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity. Communications on Pure and Applied Analysis, 2021, 20 (2) : 817-834. doi: 10.3934/cpaa.2020292
[11]	Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity. Electronic Research Archive, 2021, 29 (5) : 3281-3295. doi: 10.3934/era.2021038
[12]	Xing Liu, Yijing Sun. Multiple positive solutions for Kirchhoff type problems with singularity. Communications on Pure and Applied Analysis, 2013, 12 (2) : 721-733. doi: 10.3934/cpaa.2013.12.721
[13]	Zhongyuan Liu. Nodal Bubble-Tower Solutions for a semilinear elliptic problem with competing powers. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5299-5317. doi: 10.3934/dcds.2017230
[14]	Nemat Nyamoradi, Kaimin Teng. Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type. Communications on Pure and Applied Analysis, 2015, 14 (2) : 361-371. doi: 10.3934/cpaa.2015.14.361
[15]	Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171
[16]	Mingqi Xiang, Binlin Zhang. A critical fractional p-Kirchhoff type problem involving discontinuous nonlinearity. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 413-433. doi: 10.3934/dcdss.2019027
[17]	Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5465-5494. doi: 10.3934/dcdsb.2020354
[18]	Shu-Zhi Song, Shang-Jie Chen, Chun-Lei Tang. Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6453-6473. doi: 10.3934/dcds.2016078
[19]	Norihisa Ikoma. Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 943-966. doi: 10.3934/dcds.2015.35.943
[20]	To Fu Ma. Positive solutions for a nonlocal fourth order equation of Kirchhoff type. Conference Publications, 2007, 2007 (Special) : 694-703. doi: 10.3934/proc.2007.2007.694

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
He Zhang Haibo Chen
on Google Scholar
He Zhang Haibo Chen

2021 Impact Factor: 1.273

Tools

Article outline

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
He Zhang Haibo Chen
on Google Scholar
He Zhang Haibo Chen

[Back to Top]