EXISTENCE OF NON-TRIVIAL SOLUTIONS FOR THE KIRCHHOFF-TYPE EQUATIONS WITH FUČIK-TYPE RESONANCE AT INFINITY

Xing-Ju Chen; Zeng-Qi Ou

doi:10.11948/20200128

2021 Volume 11 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2021 > 11(2): 1006-1016

Next Article Previous Article

Xing-Ju Chen, Zeng-Qi Ou. EXISTENCE OF NON-TRIVIAL SOLUTIONS FOR THE KIRCHHOFF-TYPE EQUATIONS WITH FUČIK-TYPE RESONANCE AT INFINITY[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 1006-1016. doi: 10.11948/20200128

Citation:

Xing-Ju Chen, Zeng-Qi Ou. EXISTENCE OF NON-TRIVIAL SOLUTIONS FOR THE KIRCHHOFF-TYPE EQUATIONS WITH FUČIK-TYPE RESONANCE AT INFINITY[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 1006-1016. doi: 10.11948/20200128

EXISTENCE OF NON-TRIVIAL SOLUTIONS FOR THE KIRCHHOFF-TYPE EQUATIONS WITH FUČIK-TYPE RESONANCE AT INFINITY

Xing-Ju Chen^1, ,,
Zeng-Qi Ou^1,

1.
School of Mathematics and Statistics, Southwest University, Tiansheng Road, 400710, Beibei, Chongqing, China

Author Bio: Email address: ouzengq707@sina.com (Z. Ou)
Corresponding author: Email address: 2486601136@qq.com(X. Chen)

Abstract

Abstract

In this paper, we obtain the existence of nontrivial solutions for the Kirchhoff type equation with Fučik-type resonance at infinity by variational methods.
- Kirchhoff type equation /
- Fučik spectrum /
- mountain pass theorem /
- compactness condition
MSC: 35J20, 35J25, 35J60

FullText(HTML)

References

[1]	J. Chen and X. Tang, A non-radially symmetric solution to a class of elliptic equation with Kirchhoff term, Journal of Applied Analysis and Computation, 2019, 9, 1558-1570. doi: 10.11948/2156-907X.20180340 CrossRef Google Scholar
[2]	E. N. Dancer and Y. Du, Existence of changing sign solutions for some semilinear problems with jumping nonlinearities at zero, Proc. Roy. Soc. Edinburgh Sect. A, 1994, 124, 1165-1176. doi: 10.1017/S0308210500030171 CrossRef Google Scholar
[3]	M. Hsini, Multiplicity results for a Kirchhoff singular problem involving the fractional $p$-Laplacian, Journal of Applied Analysis and Computation. 2019, 9, 884-900. doi: 10.11948/2156-907X.20180140 CrossRef Google Scholar
[4]	G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883. Google Scholar
[5]	S. Li and Z. Zhang, Sign-changing and multiple solutions theorems for semilinear elliptic boundary value problems with jumping nonlinearities, Acta Math. Sin. (Engl. Ser. ), 2000, 16, 113-122. Google Scholar
[6]	Z. Liang, F. Li and J. Shi, Positive solutions of Kirchhoff-type non-local elliptic equation: A bifurcation approach, Proc. Roy. Soc. Edinburgh Sect. A, 2017, 147, 875-894. doi: 10.1017/S0308210516000378 CrossRef Google Scholar
[7]	F. Li, T. Rong and Z. Liang, Fučik spectrum for the Kirchhoff-type problem and applications, Nonlinear Anal., 2019, 182, 280-302. doi: 10.1016/j.na.2018.12.021 CrossRef Google Scholar
[8]	F. Li, S. X, K. X and X. Xue, Dynamic propertiles for nonlinear viscoelastic Kirchhoff-type equation with acoustic control boundary conditions Ⅱ, Journal of Applied Analysis and Computation, 2019, 9, 2318-2332. doi: 10.11948/20190085 CrossRef Google Scholar
[9]	J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Mathematics Studies, 1978, 30, 284-346. Google Scholar
[10]	P. L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 1984, 109, 33-97. doi: 10.1007/BF01205672 CrossRef Google Scholar
[11]	T. Rong, F. Li and Z. Liang, Existence of nontrivial solutions for Kirchhoff-type problems with jumping nonlinearities, Appl. Math. Lett., 2019, 95, 137-142. doi: 10.1016/j.aml.2019.03.035 CrossRef Google Scholar
[12]	J. Sun and C. Tang, Resonance problems for Kirchhoff type equations, Discrete Contin. Dyn. Syst., 2013, 33, 2139-2154. doi: 10.3934/dcds.2013.33.2139 CrossRef Google Scholar
[13]	S. Song, S. Chen and C. Tang, Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues, Discrete Contin. Dyn. Syst., 2016, 36, 6452-6473. Google Scholar
[14]	S. Song and C. Tang, Resonance problems for the $p$-Laplacian with a nonlinear boundary condition, Nonlinear Anal., 2006, 64, 2007-2021. doi: 10.1016/j.na.2005.07.035 CrossRef Google Scholar
[15]	M. Tanaka, Existence of a non-trivial solution for the $p$-Laplacian equation with Fučik type resonance at infinity Ⅱ, Nonlinear Anal. TMA, 2009, 71, 3018-3030. doi: 10.1016/j.na.2009.01.186 CrossRef Google Scholar
[16]	M. Tanaka, Existence of a non-trivial solution for the $p$-Laplacian equation with Fučik type resonance at infinity Ⅲ, Nonlinear Anal. TMA, 2010, 72, 507-526. doi: 10.1016/j.na.2009.06.096 CrossRef Google Scholar
[17]	M. Willem, Minimax Theorems, Birkhauser, Boston, 1996. Google Scholar
[18]	B. Yan, D. O'regan and R. P. Agarwal, On spectral asymptotics and bifurction for some elliptic equations of Kirchhoff-type with odd superlinear term, Journal of Applied Analysis and Computation, 2018, 8, 509-523. doi: 10.11948/2018.509 CrossRef Google Scholar
[19]	Z. Zhang and S. Li, On sign-changing and multiple solutions of the $p$-Laplacian, J. Funct. Anal., 2003, 197, 447-468. doi: 10.1016/S0022-1236(02)00103-9 CrossRef Google Scholar