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Existence of positive solutions for a class of Kirchhoff type equations in $\mathbb{R}^3$
1. | School of Mathematics and Computer Science, Hubei University of Arts and Science, Xiangyang, Hubei 441053, China |
2. | Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 |
References:
[1] |
S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475.
doi: 10.1007/BF01206962. |
[2] |
S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215.
doi: 10.1006/jfan.1996.0125. |
[3] |
C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93.
doi: 10.1016/j.camwa.2005.01.008. |
[4] |
C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbbR^N $, Nonlinear Anal., 75 (2012), 2750-2759.
doi: 10.1016/j.na.2011.11.017. |
[5] |
P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.
doi: 10.1007/BF02100605. |
[6] |
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. |
[7] |
A. Azzollini and P. d'Avenia, A. Pomponio, Multiple critical points for a class of nonlinear functionals, Ann. Mat. Pura Appl., 190 (2011), 507-523.
doi: 10.1007/s10231-010-0160-3. |
[8] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375.
doi: 10.1007/BF00250556. |
[9] |
S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées, Bull. Acad. Sci. URSS, Ser. Mat., 4 (1940), 17-26 (Izvestia Akad. Nauk SSSR). |
[10] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6 (2001), 701-730. |
[11] |
J. Q. Chen, Multiple positive solutions to a class of Kirchhoff equation on $\mathbbR^3$ with indefinite nonlinearity, Nonlinear Analysis, 96 (2014), 134-145.
doi: 10.1016/j.na.2013.11.012. |
[12] |
B. T. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl., 394 (2012), 488-495.
doi: 10.1016/j.jmaa.2012.04.025. |
[13] |
S. Cingolani and J. L. Gomez, Positive solutions of a semilinear elliptic equation on $\mathbbR^N$ with indefinite nonlinearity, Adv. Diff. Eq., 1 (1996), 773-791. |
[14] |
D. G. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in $\mathbbR^N$, Calc. Var. Partial Differential Equations, 13 (2001), 159-189.
doi: 10.1007/PL00009927. |
[15] |
G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Archive for Rational Mechanics and Analysis, 213 (2014), 931-979.
doi: 10.1007/s00205-014-0747-8. |
[16] |
Y. X. Guo and J. J. Nie, Existence and multiplicity of nontrivial solutions for $p$-Laplacian Schrödinger-Kirchhoff-type equations, J. Math. Anal. Appl., 428 (2015), 1054-1069.
doi: 10.1016/j.jmaa.2015.03.064. |
[17] |
X. M. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414.
doi: 10.1016/j.na.2008.02.021. |
[18] |
X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^3$, J. Differential Equations, 252 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[19] |
J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$, J. Math. Anal. Appl., 369 (2010), 564-574.
doi: 10.1016/j.jmaa.2010.03.059. |
[20] | |
[21] |
L. Liu and C. S. Chen, Study on existence of solutions for $p$-Kirchhoff elliptic equation in $\mathbbR^N$ with vanishing potential, Journal of Dynamical and Control Systems, 20 (2014), 575-592.
doi: 10.1007/s10883-014-9244-5. |
[22] |
W. Liu and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487.
doi: 10.1007/s12190-012-0536-1. |
[23] |
J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346. |
[24] |
P. L. Lions, The Concentration-Compactness Principle in the Calculus of Variations. The Locally Compact Case, Part I, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 1 (1984), 109-145. |
[25] |
K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221 (2006), 246-255.
doi: 10.1016/j.jde.2005.03.006. |
[26] |
S. I. Pohožaev, On a class of quasilinear hyperbolic equations, Mat. Sb. (N.S.) (Russian), 96 (1975), 152-166(in Russian). |
[27] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986.
doi: 10.1090/cbms/065. |
[28] |
J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222.
doi: 10.1016/j.na.2010.09.061. |
[29] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[30] |
X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287.
doi: 10.1016/j.nonrwa.2010.09.023. |
[31] |
Y. Yang and J. H. Zhang, Nontrivial solutions of a class of nonlocal problems via local linking theory, Appl. Math. Lett., 23 (2010), 377-380.
doi: 10.1016/j.aml.2009.11.001. |
[32] |
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. |
show all references
References:
[1] |
S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475.
doi: 10.1007/BF01206962. |
[2] |
S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215.
doi: 10.1006/jfan.1996.0125. |
[3] |
C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93.
doi: 10.1016/j.camwa.2005.01.008. |
[4] |
C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbbR^N $, Nonlinear Anal., 75 (2012), 2750-2759.
doi: 10.1016/j.na.2011.11.017. |
[5] |
P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.
doi: 10.1007/BF02100605. |
[6] |
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. |
[7] |
A. Azzollini and P. d'Avenia, A. Pomponio, Multiple critical points for a class of nonlinear functionals, Ann. Mat. Pura Appl., 190 (2011), 507-523.
doi: 10.1007/s10231-010-0160-3. |
[8] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375.
doi: 10.1007/BF00250556. |
[9] |
S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées, Bull. Acad. Sci. URSS, Ser. Mat., 4 (1940), 17-26 (Izvestia Akad. Nauk SSSR). |
[10] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6 (2001), 701-730. |
[11] |
J. Q. Chen, Multiple positive solutions to a class of Kirchhoff equation on $\mathbbR^3$ with indefinite nonlinearity, Nonlinear Analysis, 96 (2014), 134-145.
doi: 10.1016/j.na.2013.11.012. |
[12] |
B. T. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl., 394 (2012), 488-495.
doi: 10.1016/j.jmaa.2012.04.025. |
[13] |
S. Cingolani and J. L. Gomez, Positive solutions of a semilinear elliptic equation on $\mathbbR^N$ with indefinite nonlinearity, Adv. Diff. Eq., 1 (1996), 773-791. |
[14] |
D. G. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in $\mathbbR^N$, Calc. Var. Partial Differential Equations, 13 (2001), 159-189.
doi: 10.1007/PL00009927. |
[15] |
G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Archive for Rational Mechanics and Analysis, 213 (2014), 931-979.
doi: 10.1007/s00205-014-0747-8. |
[16] |
Y. X. Guo and J. J. Nie, Existence and multiplicity of nontrivial solutions for $p$-Laplacian Schrödinger-Kirchhoff-type equations, J. Math. Anal. Appl., 428 (2015), 1054-1069.
doi: 10.1016/j.jmaa.2015.03.064. |
[17] |
X. M. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414.
doi: 10.1016/j.na.2008.02.021. |
[18] |
X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^3$, J. Differential Equations, 252 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[19] |
J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$, J. Math. Anal. Appl., 369 (2010), 564-574.
doi: 10.1016/j.jmaa.2010.03.059. |
[20] | |
[21] |
L. Liu and C. S. Chen, Study on existence of solutions for $p$-Kirchhoff elliptic equation in $\mathbbR^N$ with vanishing potential, Journal of Dynamical and Control Systems, 20 (2014), 575-592.
doi: 10.1007/s10883-014-9244-5. |
[22] |
W. Liu and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487.
doi: 10.1007/s12190-012-0536-1. |
[23] |
J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346. |
[24] |
P. L. Lions, The Concentration-Compactness Principle in the Calculus of Variations. The Locally Compact Case, Part I, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 1 (1984), 109-145. |
[25] |
K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221 (2006), 246-255.
doi: 10.1016/j.jde.2005.03.006. |
[26] |
S. I. Pohožaev, On a class of quasilinear hyperbolic equations, Mat. Sb. (N.S.) (Russian), 96 (1975), 152-166(in Russian). |
[27] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986.
doi: 10.1090/cbms/065. |
[28] |
J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222.
doi: 10.1016/j.na.2010.09.061. |
[29] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[30] |
X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287.
doi: 10.1016/j.nonrwa.2010.09.023. |
[31] |
Y. Yang and J. H. Zhang, Nontrivial solutions of a class of nonlocal problems via local linking theory, Appl. Math. Lett., 23 (2010), 377-380.
doi: 10.1016/j.aml.2009.11.001. |
[32] |
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. |
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