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Existence and stability of periodic solutions for relativistic singular equations
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 |
$ -\left( a+b\int_{\Omega }{|}\nabla u{{|}.{2}}dx \right)\vartriangle u=\lambda u+|u{{|}.{2}}u,\ \ u\in H_{0}.{1}(\Omega ), $ |
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] | |
[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] | |
[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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