ON SPECTRAL ASYMPTOTICS AND BIFURCATION FOR SOME ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE WITH ODD SUPERLINEAR TERM

Baoqiang Yan; Donal O'Regan; Ravi P. Agarwal

doi:10.11948/2018.509

2018 Volume 8 Issue 2

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Baoqiang Yan, Donal O'Regan, Ravi P. Agarwal. ON SPECTRAL ASYMPTOTICS AND BIFURCATION FOR SOME ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE WITH ODD SUPERLINEAR TERM[J]. Journal of Applied Analysis & Computation, 2018, 8(2): 509-523. doi: 10.11948/2018.509

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Baoqiang Yan, Donal O'Regan, Ravi P. Agarwal. ON SPECTRAL ASYMPTOTICS AND BIFURCATION FOR SOME ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE WITH ODD SUPERLINEAR TERM[J]. Journal of Applied Analysis & Computation, 2018, 8(2): 509-523. doi: 10.11948/2018.509

ON SPECTRAL ASYMPTOTICS AND BIFURCATION FOR SOME ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE WITH ODD SUPERLINEAR TERM

1 School of Mathematical Sciences, Shandong Normal University, Jinan, 250014, China;
2 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland;
3 Department of Mathematics, Texas A and M University-Kingsville, Texas 78363, USA

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Abstract

Abstract

In this paper, from estimating the eigenvalues for Kirchhoff elliptic equations, we obtain spectral asymptotics and bifurcation concerning the eigenvalues of some related elliptic linear problem.
- Kirchhoff elliptic equations /
- Liusternik-Schnirelmann(LS)theory /
- eigenvalue /
- eigenfunction
MSC: 35P20;35P30;35J60

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References

[1]	C. O. Alves and F. J. S. A. Corrêa, A sub-supersolution approach for a quasilinear Kirchhoff equation, Journal of Mathematical Physics, 2015, 56, 051501. Doi:10.1063/1.4919670. Google Scholar
[2]	C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of kirchhoff-type, Computers and Mathematics with Applications, 2005, 49, 85-93. Google Scholar
[3]	H. Amann, Lusternik-Schnirelman theory and non-linear Eigenvalue Problems, Math. Ann., 1972, 199, 55-72. Google Scholar
[4]	G. M. Bisci, V. Radulescu and R. Servadei, Variational methods for nonlocal fractional problems, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2016. Google Scholar
[5]	G. M. Bisci and V. Radulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations, 2015, 54(3), 2985-3008. Google Scholar
[6]	G. M. Bisci and V. Radulescu, A sharp eigenvalue theorem for fractional elliptic equations, Israel J. Math., 2017, 219(1), 331-351. Google Scholar
[7]	J. Bruning, Zur abschatzung der spektralfunction elliptischer operatoren, Math. Z., 1974, 137, 75-85. Google Scholar
[8]	B. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 2009, 71, 4883-4892. Google Scholar
[9]	R. Chiappinelli, Remarks on bifurcation for elliptic operator with odd nonlinearity, Israel Journal of Mathematics, 1989, 65(3), 285-293. Google Scholar
[10]	R. Chiappinelli, On spectral asymptotics and bifurcation for elliptic operators with odd superlinearity term, Nonlinear Analysis:Theory, Methods and Applications, 1989, 13(7), 871-878. Google Scholar
[11]	Y. Deng, S. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in R³, J. Functional Analysis, 2015, 269, 3500-3527. Google Scholar
[12]	X. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 2009, 70, 1407-1414. Google Scholar
[13]	J. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in RN, J. Math. Anal. Appl., 2010, 368(2), 564-574. Google Scholar
[14]	G. Kirchhoff, Vorlesungenuber mechanik, Teubner, Leipzig, Germany, 1883. Google Scholar
[15]	X. Li and J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica, 2016, 64, 63-69. Google Scholar
[16]	X. Li and S. Song, Stabilization of Delay Systems:Delay-dependent Impulsive Control, IEEE Transactions on Automatic Control, 2017, 62(1), 406-411. Google Scholar
[17]	Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 2012, 253(7), 2285-2294. Google Scholar
[18]	Z. Liang, F. Li and J. Shi, Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior, Ann. I. H. Poincaré, 2014, 31, 155-167. Google Scholar
[19]	T. F. Ma and J. E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 2003, 16, 243-248. Google Scholar
[20]	K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ., 2006, 221, 246-255. Google Scholar
[21]	G. Prodi, Eigenvalues of non-linear problems, Cremonese, Roma, 1974. Google Scholar
[22]	P. Pucci, M. Xiang and B. Zhang, Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations, Adv. Nonlinear Anal., 2016, 5(1), 27-55. Google Scholar
[23]	V. Radulescu, M. Xiang and B. Zhang, Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-Laplacian, Nonlinearity, 2016, 29(10), 3186-3205. Google Scholar
[24]	T. Shibata, Asymptotic properties of variational eigenvalues for semilinear elliptic operators, Boll. Un. Mat. Ital., 1988, 7(2B), 411-426. Google Scholar
[25]	T. Shibata, Precise asymptotic formulas for semilinear eigenvalue problems, Ann. Henri Poincaré, 2001, 2, 713-732. Google Scholar
[26]	T. Shibata, Precise spectral asymptotics for nonlinear Sturm-Liouville problems, J. Differential Equations, 2002, 180, 374-394. Google Scholar
[27]	T. Shibata, Global behavior of the branch of positive solutions to a logistic equation of population dynamics, Proc. Amer. Math. Soc., 2008, 136(7), 2547-2554. Google Scholar
[28]	W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equations, 2015, 259, 1256-1274. Google Scholar
[29]	B. Yan and D. Wang, The multiplicity of positive solutions for a class of nonlocal elliptic problemJournal of Mathematical Analysis and Applications, 2016, 442(1), 72-102. Google Scholar
[30]	X. Zhang, B. Zhang and M. Xiang, Ground states for fractional Schrödinger equations involving a critical nonlinearity, Adv. Nonlinear Anal., 2016, 5(3), 293-314. Google Scholar