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Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type
1. | School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205 Hunan, China |
2. | School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083 |
3. | School of Mathematics and Statistics, Qujing Normal University, Qujing, 655011 Yunnan, China |
References:
[1] |
C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153-1166.
doi: 10.1007/s00033-013-0376-3. |
[2] |
J. M. Ball, Initial-boundary value for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90. |
[3] |
H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. |
[4] |
T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$, Commun. Partial Differ. Equ., 20 (1995), 1725-1741.
doi: 10.1080/03605309508821149. |
[5] |
T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. I. H. Poincaré Anal, Non Linéaire, 22 (2005), 259-281.
doi: 10.1016/j.anihpc.2004.07.005. |
[6] |
T. Bartch, 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. |
[7] |
T. Bartsch, Z. Q. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations, in Handbook of Differential Equations-Stationary Partial Differential Equations (M. Chipot and P. Quittner eds.), vol. 2, Elsevier, 2005, pp. 1-55, Chapter 1.
doi: 10.1016/S1874-5733(05)80009-9. |
[8] |
B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 71 (2009), 4883-4892.
doi: 10.1016/j.na.2009.03.065. |
[9] |
B. T. Cheng, X. Wu and J. Liu, Multiple solutions for a class of Kirchhoff-type problems with concave nonlinearity, NoDEA Nonl. Diff. Equa. Appl., 19 (2012), 521-537.
doi: 10.1007/s00030-011-0141-2. |
[10] |
B. T. Cheng and X. H. Tang, Infinitely many large energy solutions for Schrödinger-Kirchhoff type problem in $\mathbbR^N$, J. Nonlinear Sci. Appl., 9 (2016), 652-660. |
[11] |
X. J. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2965-2992.
doi: 10.1016/j.jde.2014.01.027. |
[12] |
E. N. Dancer and Y. H. Du, On sign-changing solutions of certain semilinear elliptic problems, Appl. Anal., 56 (1995), 193-206.
doi: 10.1080/00036819508840321. |
[13] |
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. |
[14] |
X. M. He and W. M. Zou, Existen and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^{3}$, J. Differential Equations, 2 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[15] |
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. |
[16] |
Z. L. Liu and J. X. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172 (2001), 257-299.
doi: 10.1006/jdeq.2000.3867. |
[17] |
J. Liu, S. X. Chen and X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $\mathbbR^N$, J. Math. Anal. Appl., 395 (2012), 608-615.
doi: 10.1016/j.jmaa.2012.05.063. |
[18] |
H. L. Liu and H. B. Chen, Haibo Least energy nodal solution for quasilinear biharmonic equations with critical exponent in $\mathbbR^N$, Appl. Math. Lett., 48 (2015), 85-90.
doi: 10.1016/j.aml.2015.03.002. |
[19] |
T. F. Ma, Positive solutions for a nonlocal fourth order equation of Kirchhoff type, Disc. Contin. Dyn. Syst., Supplement (2007), 694-703. |
[20] |
T. F. Ma, Existence results for a model of nonlinear beam on elastic bearings, Appl. Math. Lett., 13 (2000), 11-15.
doi: 10.1016/S0893-9659(00)00026-4. |
[21] |
T. F. Ma, Existence results and numerical solutions for a beam equation with nonlinear boundary conditions, Appl. Numer. Math., 47 (2003), 189-196.
doi: 10.1016/S0168-9274(03)00065-5. |
[22] |
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. |
[23] |
P. J. McKenna and W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), 703-715.
doi: 10.1137/0150041. |
[24] |
J. J. Nie, Existence and multiplicity of nontrivial solutions for a class of Schrödinger-Kirchhoff-type equations, J. Math. Anal. Appl., 417 (2014), 65-79.
doi: 10.1016/j.jmaa.2014.03.027. |
[25] |
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. |
[26] |
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. |
[27] |
W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. |
[28] |
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. |
[29] |
F. L. Wang, M. Avci and Y. K. An, Existence of solutions for fourth-order elliptic equations of Kirchhoff type, J. Math. Anal. Appl., 409 (2014), 140-146.
doi: 10.1016/j.jmaa.2013.07.003. |
[30] |
M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[31] |
L. P. Xu and H. B. Chen, Nontrivial solutions for Kirchhoff-type problems with a parameter, J. Math. Anal. Appl., 433 (2016), 455-472.
doi: 10.1016/j.jmaa.2015.07.035. |
[32] |
L. P. Xu and H. B. Chen, Multiplicity results for fourth order elliptic equations of Kirchhoff-type, Acta Math. Sci. Ser. B, 35 (2015), 1067-1076.
doi: 10.1016/S0252-9602(15)30040-0. |
[33] |
L. P. Xu and H. B. Chen, Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type via genus theory, Bound. Value Probl., 2014 (2014), 212.
doi: 10.1186/s13661-014-0212-5. |
[34] |
Y. L. Yin and X. Wu, High energy solutions and nontrivial solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 375 (2011), 699-705.
doi: 10.1016/j.jmaa.2010.10.019. |
[35] |
Y. W. Ye and C. L. Tang, Infinitely many solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 394 (2012), 841-854.
doi: 10.1016/j.jmaa.2012.04.041. |
[36] |
Y. W. Ye and C. L. Tang, Existence and multiplicity of solutions for fourth-order elliptic equations in $\mathbbR^N$, J. Math. Anal. Appl., 406 (2013), 335-351.
doi: 10.1016/j.jmaa.2013.04.079. |
[37] |
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. |
[38] |
J. Zhang, X. H. Tang and W. Zhang, Existence of multiple solutions of Kirchhoff type equation with sign-changing potential, Appl. Math. Compu., 242 (2014), 491-499.
doi: 10.1016/j.amc.2014.05.070. |
[39] |
W. Zhang, X. H. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with general potentials, J. Math. Anal. Appl., 407 (2013), 359-368.
doi: 10.1016/j.jmaa.2013.05.044. |
[40] |
W. Zhang, X. H. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with sign-changing potential, Taiwanese J. Math., 18 (2014), 645-659.
doi: 10.11650/tjm.18.2014.3584. |
[41] |
W. Zhang, X. H. Tang and J. Zhang, Ground states for a class of asymptotically linear fourthorder elliptic equations, Appl. Anal., 94 (2015), 2168-2174.
doi: 10.1080/00036811.2014.979807. |
[42] |
W. Zhang, X. H. Tang and J. Zhang, Existence and concentration of solutions for sublinear fourth-order elliptic equations, Electron. J. Diff. Equ., 2015 (2015), 1-9. |
show all references
References:
[1] |
C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153-1166.
doi: 10.1007/s00033-013-0376-3. |
[2] |
J. M. Ball, Initial-boundary value for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90. |
[3] |
H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. |
[4] |
T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$, Commun. Partial Differ. Equ., 20 (1995), 1725-1741.
doi: 10.1080/03605309508821149. |
[5] |
T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. I. H. Poincaré Anal, Non Linéaire, 22 (2005), 259-281.
doi: 10.1016/j.anihpc.2004.07.005. |
[6] |
T. Bartch, 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. |
[7] |
T. Bartsch, Z. Q. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations, in Handbook of Differential Equations-Stationary Partial Differential Equations (M. Chipot and P. Quittner eds.), vol. 2, Elsevier, 2005, pp. 1-55, Chapter 1.
doi: 10.1016/S1874-5733(05)80009-9. |
[8] |
B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 71 (2009), 4883-4892.
doi: 10.1016/j.na.2009.03.065. |
[9] |
B. T. Cheng, X. Wu and J. Liu, Multiple solutions for a class of Kirchhoff-type problems with concave nonlinearity, NoDEA Nonl. Diff. Equa. Appl., 19 (2012), 521-537.
doi: 10.1007/s00030-011-0141-2. |
[10] |
B. T. Cheng and X. H. Tang, Infinitely many large energy solutions for Schrödinger-Kirchhoff type problem in $\mathbbR^N$, J. Nonlinear Sci. Appl., 9 (2016), 652-660. |
[11] |
X. J. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2965-2992.
doi: 10.1016/j.jde.2014.01.027. |
[12] |
E. N. Dancer and Y. H. Du, On sign-changing solutions of certain semilinear elliptic problems, Appl. Anal., 56 (1995), 193-206.
doi: 10.1080/00036819508840321. |
[13] |
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. |
[14] |
X. M. He and W. M. Zou, Existen and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^{3}$, J. Differential Equations, 2 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[15] |
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. |
[16] |
Z. L. Liu and J. X. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172 (2001), 257-299.
doi: 10.1006/jdeq.2000.3867. |
[17] |
J. Liu, S. X. Chen and X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $\mathbbR^N$, J. Math. Anal. Appl., 395 (2012), 608-615.
doi: 10.1016/j.jmaa.2012.05.063. |
[18] |
H. L. Liu and H. B. Chen, Haibo Least energy nodal solution for quasilinear biharmonic equations with critical exponent in $\mathbbR^N$, Appl. Math. Lett., 48 (2015), 85-90.
doi: 10.1016/j.aml.2015.03.002. |
[19] |
T. F. Ma, Positive solutions for a nonlocal fourth order equation of Kirchhoff type, Disc. Contin. Dyn. Syst., Supplement (2007), 694-703. |
[20] |
T. F. Ma, Existence results for a model of nonlinear beam on elastic bearings, Appl. Math. Lett., 13 (2000), 11-15.
doi: 10.1016/S0893-9659(00)00026-4. |
[21] |
T. F. Ma, Existence results and numerical solutions for a beam equation with nonlinear boundary conditions, Appl. Numer. Math., 47 (2003), 189-196.
doi: 10.1016/S0168-9274(03)00065-5. |
[22] |
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. |
[23] |
P. J. McKenna and W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), 703-715.
doi: 10.1137/0150041. |
[24] |
J. J. Nie, Existence and multiplicity of nontrivial solutions for a class of Schrödinger-Kirchhoff-type equations, J. Math. Anal. Appl., 417 (2014), 65-79.
doi: 10.1016/j.jmaa.2014.03.027. |
[25] |
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. |
[26] |
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. |
[27] |
W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. |
[28] |
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. |
[29] |
F. L. Wang, M. Avci and Y. K. An, Existence of solutions for fourth-order elliptic equations of Kirchhoff type, J. Math. Anal. Appl., 409 (2014), 140-146.
doi: 10.1016/j.jmaa.2013.07.003. |
[30] |
M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[31] |
L. P. Xu and H. B. Chen, Nontrivial solutions for Kirchhoff-type problems with a parameter, J. Math. Anal. Appl., 433 (2016), 455-472.
doi: 10.1016/j.jmaa.2015.07.035. |
[32] |
L. P. Xu and H. B. Chen, Multiplicity results for fourth order elliptic equations of Kirchhoff-type, Acta Math. Sci. Ser. B, 35 (2015), 1067-1076.
doi: 10.1016/S0252-9602(15)30040-0. |
[33] |
L. P. Xu and H. B. Chen, Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type via genus theory, Bound. Value Probl., 2014 (2014), 212.
doi: 10.1186/s13661-014-0212-5. |
[34] |
Y. L. Yin and X. Wu, High energy solutions and nontrivial solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 375 (2011), 699-705.
doi: 10.1016/j.jmaa.2010.10.019. |
[35] |
Y. W. Ye and C. L. Tang, Infinitely many solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 394 (2012), 841-854.
doi: 10.1016/j.jmaa.2012.04.041. |
[36] |
Y. W. Ye and C. L. Tang, Existence and multiplicity of solutions for fourth-order elliptic equations in $\mathbbR^N$, J. Math. Anal. Appl., 406 (2013), 335-351.
doi: 10.1016/j.jmaa.2013.04.079. |
[37] |
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. |
[38] |
J. Zhang, X. H. Tang and W. Zhang, Existence of multiple solutions of Kirchhoff type equation with sign-changing potential, Appl. Math. Compu., 242 (2014), 491-499.
doi: 10.1016/j.amc.2014.05.070. |
[39] |
W. Zhang, X. H. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with general potentials, J. Math. Anal. Appl., 407 (2013), 359-368.
doi: 10.1016/j.jmaa.2013.05.044. |
[40] |
W. Zhang, X. H. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with sign-changing potential, Taiwanese J. Math., 18 (2014), 645-659.
doi: 10.11650/tjm.18.2014.3584. |
[41] |
W. Zhang, X. H. Tang and J. Zhang, Ground states for a class of asymptotically linear fourthorder elliptic equations, Appl. Anal., 94 (2015), 2168-2174.
doi: 10.1080/00036811.2014.979807. |
[42] |
W. Zhang, X. H. Tang and J. Zhang, Existence and concentration of solutions for sublinear fourth-order elliptic equations, Electron. J. Diff. Equ., 2015 (2015), 1-9. |
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