March  2017, 16(2): 611-628. doi: 10.3934/cpaa.2017030

The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

2. 

School of History Culture and Ethnology, Southwest University, Chongqing 400715, China

* Corresponding author

Received  July 2016 Revised  September 2016 Published  January 2017

Fund Project: supported by National Natural Science Foundation of China(No. 11471267); the Fundamental Research Funds for the Central Universities (No. SWU1109075)

In this paper, we investigate the existence and nonexistence of ground state nodal solutions to a class of Kirchhoff type problems
$ -\left( a+b\int_{\Omega }{|}\nabla u{{|}.{2}}dx \right)\vartriangle u=\lambda u+|u{{|}.{2}}u,\ \ u\in H_{0}.{1}(\Omega ), $
where $a, b>0$, $\lambda < a\lambda_1$, $\lambda_1$ is the principal eigenvalue of $(-\triangle, H_0.{1}(\Omega))$. With the help of the Nehari manifold, we obtain that there is $\Lambda>0$ such that the Kirchhoff type problem possesses at least one ground state nodal solution $u_b$ for all $0 < b < \Lambda$ and $\lambda < a\lambda_1$ and prove that its energy is strictly larger than twice that of ground state solutions. Moreover, we give a convergence property of $u_b$ as $b\searrow 0$. Besides, we firstly establish the nonexistence result of nodal solutions for all $b\geq\Lambda$. This paper can be regarded as the extension and complementary work of W. Shuai (2015)[21], X.H. Tang and B.T. Cheng (2016)[22].
Citation: Xiao-Jing Zhong, Chun-Lei Tang. The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem. Communications on Pure & Applied Analysis, 2017, 16 (2) : 611-628. doi: 10.3934/cpaa.2017030
References:
[1]

A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330.   Google Scholar

[2]

T. BartschZ. L. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Comm. Partial Differential Equations, 29 (2004), 25-42.  doi: 10.1142/9789812704283_0027.  Google Scholar

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T. Bartsch and T. Weth, Three nodal solutions of singular 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.  Google Scholar

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T. BartschT. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18.  doi: 10.1007/BF02787822.  Google Scholar

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K. J. Brown and T. F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function, J. Math. Anal. Appl., 337 (2008), 1326-1336.  doi: 10.1016/j.jmaa.2007.04.064.  Google Scholar

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K. J. Brown and Y. P. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

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Y. B. DengS. J. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}$3, Journal of Functional Analysis, 269 (2015), 3500-3527.  doi: 10.1016/j.jfa.2015.09.012.  Google Scholar

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G. M. Figueiredo and R. G. 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

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Y. HeG. B. Li and S. J. 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.  Google Scholar

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X. M. He and W. M. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura. Appl., 193 (2014), 473-500.  doi: 10.1007/s10231-012-0286-6.  Google Scholar

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

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C. Y. LeiJ. F. Liao and C. L. Tang, Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 421 (2015), 521-538.  doi: 10.1016/j.jmaa.2014.07.031.  Google Scholar

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S. H. Liang and J. H. Zhang, Existence of solutions for Kirchhoff type problems with critical nonlinearity in $\mathbb{R}$3, Nonlinear Anal. Real World Appl., 17 (2014), 126-136.  doi: 10.1016/j.nonrwa.2013.10.011.  Google Scholar

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J. L. Lions, On some questions in boundary value problems of mathematical physics, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proceedings of International Symposium, Inst. Mat. , Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, in: North-Holland Math. Stud. , vol. 30, North-Holland, Amsterdam, 1978, pp. 284–346.  Google Scholar

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J. LiuJ. F. Liao and C. L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in $\mathbb{R}$N, J. Math. Anal. Appl., 429 (2015), 1153-1172.  doi: 10.1016/j.jmaa.2015.04.066.  Google Scholar

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S. 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

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A. M. Mao and S. X. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239-243.  doi: 10.1016/j.jmaa.2011.05.021.  Google Scholar

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A. M. Mao and Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287.  doi: 10.1016/j.na.2008.02.011.  Google Scholar

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D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 257 (2014), 1168-1193.  doi: 10.1016/j.jde.2014.05.002.  Google Scholar

[21]

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

[22]

X. H. Tang and B. T. 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

[23]

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. Differential Equations, 253 (2012), 2314-2351.  doi: 10.1016/j.jde.2012.05.023.  Google Scholar

[24]

Z. P. Wang and H. S. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}$3, Calc. Var., 52 (2015), 927-943.  doi: 10.1007/s00526-014-0738-5.  Google Scholar

[25]

L. P. Xu and H. B. Chen, Sign-changing solutions to Schrödinger-Kirchhoff-type equations with critical exponent, Advances in Difference Equations, 1 (2016), 1-14.  doi: 10.1186/s13662-016-0828-0.  Google Scholar

[26]

H. Zhang and F. B. Zhang, Ground states for the nonlinear Kirchhoff type problems, J. Math. Anal. Appl., 423 (2015), 1671-1692.  doi: 10.1016/j.jmaa.2014.10.062.  Google Scholar

[27]

J. Zhang, On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 428 (2015), 387-404.  doi: 10.1016/j.jmaa.2015.03.032.  Google Scholar

[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.  Google Scholar

[29]

W. M. Zou, Sign-Changing Critical Point Theory, Spring, New York, 2008. Google Scholar

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.   Google Scholar

[2]

T. BartschZ. L. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Comm. Partial Differential Equations, 29 (2004), 25-42.  doi: 10.1142/9789812704283_0027.  Google Scholar

[3]

T. Bartsch and T. Weth, Three nodal solutions of singular 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.  Google Scholar

[4]

T. BartschT. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18.  doi: 10.1007/BF02787822.  Google Scholar

[5]

S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées partielles, Bull. Acad. Sci. URSS. Sér. (Izvestia Akad. Nauk SSSR), 4 (1940), 17-26.   Google Scholar

[6]

K. J. Brown and T. F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function, J. Math. Anal. Appl., 337 (2008), 1326-1336.  doi: 10.1016/j.jmaa.2007.04.064.  Google Scholar

[7]

K. J. Brown and Y. P. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

[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, Journal of Functional Analysis, 269 (2015), 3500-3527.  doi: 10.1016/j.jfa.2015.09.012.  Google Scholar

[9]

G. M. Figueiredo and R. G. 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

[10]

Y. HeG. B. Li and S. J. 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.  Google Scholar

[11]

X. M. He and W. M. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura. Appl., 193 (2014), 473-500.  doi: 10.1007/s10231-012-0286-6.  Google Scholar

[12]

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

[13]

C. Y. LeiJ. F. Liao and C. L. Tang, Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 421 (2015), 521-538.  doi: 10.1016/j.jmaa.2014.07.031.  Google Scholar

[14]

S. H. Liang and J. H. Zhang, Existence of solutions for Kirchhoff type problems with critical nonlinearity in $\mathbb{R}$3, Nonlinear Anal. Real World Appl., 17 (2014), 126-136.  doi: 10.1016/j.nonrwa.2013.10.011.  Google Scholar

[15]

J. L. Lions, On some questions in boundary value problems of mathematical physics, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proceedings of International Symposium, Inst. Mat. , Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, in: North-Holland Math. Stud. , vol. 30, North-Holland, Amsterdam, 1978, pp. 284–346.  Google Scholar

[16]

J. LiuJ. F. Liao and C. L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in $\mathbb{R}$N, J. Math. Anal. Appl., 429 (2015), 1153-1172.  doi: 10.1016/j.jmaa.2015.04.066.  Google Scholar

[17]

S. 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

[18]

A. M. Mao and S. X. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239-243.  doi: 10.1016/j.jmaa.2011.05.021.  Google Scholar

[19]

A. M. Mao and Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287.  doi: 10.1016/j.na.2008.02.011.  Google Scholar

[20]

D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 257 (2014), 1168-1193.  doi: 10.1016/j.jde.2014.05.002.  Google Scholar

[21]

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

[22]

X. H. Tang and B. T. 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

[23]

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. Differential Equations, 253 (2012), 2314-2351.  doi: 10.1016/j.jde.2012.05.023.  Google Scholar

[24]

Z. P. Wang and H. S. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}$3, Calc. Var., 52 (2015), 927-943.  doi: 10.1007/s00526-014-0738-5.  Google Scholar

[25]

L. P. Xu and H. B. Chen, Sign-changing solutions to Schrödinger-Kirchhoff-type equations with critical exponent, Advances in Difference Equations, 1 (2016), 1-14.  doi: 10.1186/s13662-016-0828-0.  Google Scholar

[26]

H. Zhang and F. B. Zhang, Ground states for the nonlinear Kirchhoff type problems, J. Math. Anal. Appl., 423 (2015), 1671-1692.  doi: 10.1016/j.jmaa.2014.10.062.  Google Scholar

[27]

J. Zhang, On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 428 (2015), 387-404.  doi: 10.1016/j.jmaa.2015.03.032.  Google Scholar

[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.  Google Scholar

[29]

W. M. Zou, Sign-Changing Critical Point Theory, Spring, New York, 2008. Google Scholar

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