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Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities
1. | School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China |
2. | School of Mathematics and Information, China West Normal University, Nanchong, Sichuan, 637002, China |
In this paper, we are interested in looking for multiple solutions for a class of Kirchhoff type problems with concave-convex nonlinearities. Under the combined effect of coefficient functions of concave-convex nonlinearities, by the Nehari method, we obtain two solutions, and one of them is a ground state solution. Under some stronger conditions, we point that the two solutions are positive solutions by the strong maximum principle.
References:
[1] |
C. O. Alves, F. J. S. A. Corrȇa 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. |
[2] |
A. Ambrosetti, H. Brézis and G. Cerami,
Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[3] |
G. Anello,
A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problem, J. Math. Anal. Appl., 373 (2011), 248-251.
doi: 10.1016/j.jmaa.2010.07.019. |
[4] |
C. Y. Chen, Y. C. Kuo and T. F. Wu,
The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908.
doi: 10.1016/j.jde.2010.11.017. |
[5] |
B. T. Cheng, Xian Wu and Liu Jun,
Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 521-537.
doi: 10.1007/s00030-011-0141-2. |
[6] |
Y. B. Deng, S. 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. |
[7] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[8] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order Springer-Verlag, Berlin, 2001. |
[9] |
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. |
[10] |
G. B. Li and H. Y. Ye,
Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^3$, J. Differential Equations, 257 (2014), 566-600.
doi: 10.1016/j.jde.2014.04.011. |
[11] |
Y. H. Li, F. Y. Li and J. P. Shi,
Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285-2294.
doi: 10.1016/j.jde.2012.05.017. |
[12] |
J. F. Liao, P. Zhang, J. Liu and C. L. Tang,
Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity, J. Math. Anal. Appl., 430 (2015), 1124-1148.
doi: 10.1016/j.jmaa.2015.05.038. |
[13] |
J. F. Liao, P. Zhang, J. Liu and C. L. Tang,
Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1959-1974.
doi: 10.3934/dcdss.2016080. |
[14] |
J. F. Liao, X. F. Ke, C. Y. Lei and C. L. Tang,
A uniqueness result for Kirchhoff type problems with singularity, Appl. Math. Lett., 59 (2016), 24-30.
doi: 10.1016/j.aml.2016.03.001. |
[15] |
Z. S. Liu and S. J. Guo,
Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent, Z. Angew. Math. Phys., 66 (2015), 747-769.
doi: 10.1007/s00033-014-0431-8. |
[16] |
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. |
[17] |
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. |
[18] |
D. Naimen,
Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 885-914.
doi: 10.1007/s00030-014-0271-4. |
[19] |
J. J. Nie and X. Wu,
Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479.
doi: 10.1016/j.na.2012.01.004. |
[20] |
P. Pucci, M. Q. Xiang and B. L. Zhang,
Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 54 (2015), 1-22.
doi: 10.1007/s00526-015-0883-5. |
[21] |
W. Rudin,
Real and Complex Analysis McGraw-Hill, New York, London etc. 1966. |
[22] |
J. J. Sun and C. L. Tang,
Existence and multipicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222.
doi: 10.1016/j.na.2010.09.061. |
[23] |
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. |
[24] |
L. Wei 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. |
[25] |
X. Wu,
Existence of nontrivial solutions and high energy solutions for Schrodinger-Kirchhoff-type equations in $\mathbb{R}^{3}$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287.
doi: 10.1016/j.nonrwa.2010.09.023. |
[26] |
M. Q. Xiang, B. L. Zhang and V. D. Rǎdulescu,
Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-laplacian, Nonlinearity, 29 (2016), 3186-3205.
doi: 10.1088/0951-7715/29/10/3186. |
[27] |
Q. L. Xie, S. W. Ma and X. Zhang,
Bound state solutions of Kirchhoff type problems with critical exponent, J. Differential Equations, 261 (2016), 890-924.
doi: 10.1016/j.jde.2016.03.028. |
[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. |
show all references
References:
[1] |
C. O. Alves, F. J. S. A. Corrȇa 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. |
[2] |
A. Ambrosetti, H. Brézis and G. Cerami,
Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[3] |
G. Anello,
A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problem, J. Math. Anal. Appl., 373 (2011), 248-251.
doi: 10.1016/j.jmaa.2010.07.019. |
[4] |
C. Y. Chen, Y. C. Kuo and T. F. Wu,
The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908.
doi: 10.1016/j.jde.2010.11.017. |
[5] |
B. T. Cheng, Xian Wu and Liu Jun,
Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 521-537.
doi: 10.1007/s00030-011-0141-2. |
[6] |
Y. B. Deng, S. 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. |
[7] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[8] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order Springer-Verlag, Berlin, 2001. |
[9] |
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. |
[10] |
G. B. Li and H. Y. Ye,
Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^3$, J. Differential Equations, 257 (2014), 566-600.
doi: 10.1016/j.jde.2014.04.011. |
[11] |
Y. H. Li, F. Y. Li and J. P. Shi,
Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285-2294.
doi: 10.1016/j.jde.2012.05.017. |
[12] |
J. F. Liao, P. Zhang, J. Liu and C. L. Tang,
Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity, J. Math. Anal. Appl., 430 (2015), 1124-1148.
doi: 10.1016/j.jmaa.2015.05.038. |
[13] |
J. F. Liao, P. Zhang, J. Liu and C. L. Tang,
Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1959-1974.
doi: 10.3934/dcdss.2016080. |
[14] |
J. F. Liao, X. F. Ke, C. Y. Lei and C. L. Tang,
A uniqueness result for Kirchhoff type problems with singularity, Appl. Math. Lett., 59 (2016), 24-30.
doi: 10.1016/j.aml.2016.03.001. |
[15] |
Z. S. Liu and S. J. Guo,
Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent, Z. Angew. Math. Phys., 66 (2015), 747-769.
doi: 10.1007/s00033-014-0431-8. |
[16] |
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. |
[17] |
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. |
[18] |
D. Naimen,
Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 885-914.
doi: 10.1007/s00030-014-0271-4. |
[19] |
J. J. Nie and X. Wu,
Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479.
doi: 10.1016/j.na.2012.01.004. |
[20] |
P. Pucci, M. Q. Xiang and B. L. Zhang,
Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 54 (2015), 1-22.
doi: 10.1007/s00526-015-0883-5. |
[21] |
W. Rudin,
Real and Complex Analysis McGraw-Hill, New York, London etc. 1966. |
[22] |
J. J. Sun and C. L. Tang,
Existence and multipicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222.
doi: 10.1016/j.na.2010.09.061. |
[23] |
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. |
[24] |
L. Wei 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. |
[25] |
X. Wu,
Existence of nontrivial solutions and high energy solutions for Schrodinger-Kirchhoff-type equations in $\mathbb{R}^{3}$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287.
doi: 10.1016/j.nonrwa.2010.09.023. |
[26] |
M. Q. Xiang, B. L. Zhang and V. D. Rǎdulescu,
Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-laplacian, Nonlinearity, 29 (2016), 3186-3205.
doi: 10.1088/0951-7715/29/10/3186. |
[27] |
Q. L. Xie, S. W. Ma and X. Zhang,
Bound state solutions of Kirchhoff type problems with critical exponent, J. Differential Equations, 261 (2016), 890-924.
doi: 10.1016/j.jde.2016.03.028. |
[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. |
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