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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

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.
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. 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.

[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. 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.  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. LiuH. 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. SunT. 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. 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.

[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. 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.  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. LiuH. 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. SunT. 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.

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