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.
Citation: |
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
I. Ekeland
, On the variational principle, J. Math. Anal. Appl., 47 (1974)
, 324-353.
doi: 10.1016/0022-247X(74)90025-0.![]() ![]() ![]() |
|
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order Springer-Verlag, Berlin, 2001.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
W. Rudin,
Real and Complex Analysis McGraw-Hill, New York, London etc. 1966.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |