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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

Xiao-Jing Zhong 1,2,, and  Chun-Lei Tang 1,2,,

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].
Keywords: Kirchhoff type problem, nonlocal term, Nehari manifold, nodal solution, ground state.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35J65.
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 and 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.

[2]

T. Bartsch, Z. 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.

[3]

T. Bartsch and T. Weth, Three nodal solutions of singular perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 22 (2005), 259–281. doi: 10.1016/j.anihpc.2004.07.005.

[4]

T. Bartsch, T. 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.

[5]

S. Bernstein, Sur une classe d'´equations fonctionnelles aux d´eriv´ees partielles, Bull. Acad. Sci. URSS. S´er. (Izvestia Akad. Nauk SSSR), 4 (1940), 17–26.

[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.

[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.

[8]

Y. B. Deng, S. J. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in R3, Journal of Functional Analysis, 269 (2015), 3500–3527. doi: 10.1016/j.jfa.2015.09.012.

[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.

[10]

Y. He, G. B. Li and S. J. Peng, Concentrating bound states for Kirchhoff type problems in R3 involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (2014), 483–510. doi: 10.1515/ans-2014-0214.

[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.

[12]

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

[13]

C. Y. Lei, J. 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.

[14]

S. H. Liang and J. H. Zhang, Existence of solutions for Kirchhoff type problems with critical nonlinearity in R3, Nonlinear Anal. Real World Appl., 17 (2014), 126–136. doi: 10.1016/j.nonrwa.2013.10.011.

[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.

[16]

J. Liu, J. F. Liao and C. L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in RN, J. Math. Anal. Appl., 429 (2015), 1153–1172. doi: 10.1016/j.jmaa.2015.04.066.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

J. Wang, L. X. Tian, J. 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.

[24]

Z. P. Wang and H. S. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in R3, Calc. Var., 52 (2015), 927–943. doi: 10.1007/s00526-014-0738-5.

[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.

[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.

[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.

[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]

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

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.

[2]

T. Bartsch, Z. 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.

[3]

T. Bartsch and T. Weth, Three nodal solutions of singular perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 22 (2005), 259–281. doi: 10.1016/j.anihpc.2004.07.005.

[4]

T. Bartsch, T. 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.

[5]

S. Bernstein, Sur une classe d'´equations fonctionnelles aux d´eriv´ees partielles, Bull. Acad. Sci. URSS. S´er. (Izvestia Akad. Nauk SSSR), 4 (1940), 17–26.

[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.

[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.

[8]

Y. B. Deng, S. J. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in R3, Journal of Functional Analysis, 269 (2015), 3500–3527. doi: 10.1016/j.jfa.2015.09.012.

[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.

[10]

Y. He, G. B. Li and S. J. Peng, Concentrating bound states for Kirchhoff type problems in R3 involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (2014), 483–510. doi: 10.1515/ans-2014-0214.

[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.

[12]

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

[13]

C. Y. Lei, J. 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.

[14]

S. H. Liang and J. H. Zhang, Existence of solutions for Kirchhoff type problems with critical nonlinearity in R3, Nonlinear Anal. Real World Appl., 17 (2014), 126–136. doi: 10.1016/j.nonrwa.2013.10.011.

[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.

[16]

J. Liu, J. F. Liao and C. L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in RN, J. Math. Anal. Appl., 429 (2015), 1153–1172. doi: 10.1016/j.jmaa.2015.04.066.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

J. Wang, L. X. Tian, J. 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.

[24]

Z. P. Wang and H. S. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in R3, Calc. Var., 52 (2015), 927–943. doi: 10.1007/s00526-014-0738-5.

[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.

[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.

[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.

[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]

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

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