Abstract
In this paper, we study the existence of infinitely many nontrivial solutions for a class of nonlinear Kirchhoff-type equation
where constants \(a>0,\ b>0\), \(\Delta_{\lambda}\) is a strongly degenerate elliptic operator, and \(f\) is a function with a more general superlinear conditions or sublinear conditions.
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REFERENCES
C. T. Anh and B. K. My, ‘‘Existence of solutions to \(\Delta_{\lambda}\)-Laplace equations without the Ambrosetti–Rabinowitz condition,’’ Complex Var. Elliptic Equations 61, 137–150 (2016). https://doi.org/10.1080/17476933.2015.1068762
C. T. Anh and B. K. My, ‘‘Liouville-type theorems for elliptic inequalities involving the \(\Delta_{\lambda}\)-Laplace operator,’’ Complex Var. Elliptic Equations 61, 1002–1013 (2016). https://doi.org/10.1080/17476933.2015.1131685
J. Chen, X. Tang, and Z. Gao, ‘‘Infinitely many solutions for semilinear \(\Delta_{\lambda}\)-Laplace equations with sign-changing potential and nonlinearity,’’ Stud. Sci. Math. Hungar. 54, 536–549 (2017). https://doi.org/10.1556/012.2017.54.4.1382
B. Cheng and X. Wu, ‘‘Existence results of positive solutions of Kirchhoff type problems,’’ Nonlinear Anal. 71, 4883–4892 (2009). https://doi.org/10.1016/j.na.2009.03.065
L. Hörmander, ‘‘Hypoelliptic second order differential equations,’’ Acta Math. 119, 147–171 (1967). https://doi.org/10.1007/BF02392081
Y.-T. Jing and Z.-L. Liu, ‘‘Elliptic systems with a partially sublinear local term,’’ J. Math. Study 48, 290–305 (2015). https://doi.org/10.4208/jms.v48n3.15.07
A. E. Kogoj and E. Lanconelli, ‘‘On semilinear \(\Delta_{\lambda}\)-Laplace equation,’’ Nonlinear Anal. 75, 4637–4649 (2012). https://doi.org/10.1016/j.na.2011.10.007
A. E. Kogoj and S. Sonner, ‘‘Attractors for a class of semi-linear degenerate parabolic equations,’’ J. Evol. Equations 13, 675–691 (2013). https://doi.org/10.1007/s00028-013-0196-0
A. E. Kogoj and S. Sonner, ‘‘Attractors met X-elliptic operators,’’ Math. Anal. Appl. 420, 407–434 (2014). https://doi.org/10.1016/j.jmaa.2014.05.070
E. Lanconelli and A. E. Kogoj, ‘‘\(X\)-elliptic operators and \(X\)-control distances,’’ Ric. Mat. 49, 223–243 (2000).
P.-L. Lions, ‘‘Symétrie et compacité dans les espaces de Sobolev,’’ J. Funct. Anal. 49, 315–334 (1982). https://doi.org/10.1016/0022-1236(82)90072-6
Z. Liu and Z.-Q. Wang, ‘‘On Clark’s theorem and its applications to partially sublinear problems,’’ Ann. Inst. Henri Poincare C 32, 1015–1037 (2015). https://doi.org/10.1016/j.anihpc.2014.05.002
D. T. Luyen, ‘‘Two nontrivial solutions of boundary-value problems for semilinear \(\Delta_{\gamma}\)-differential equations,’’ Math. Notes 101, 815–823 (2017). https://doi.org/10.1134/S0001434617050078
D. T. Luyen, D. T. Huong, and L. T. H. Hanh, ‘‘Existence of infinitely many solutions for \(\Delta_{\gamma}\)-Laplace problems,’’ Math. Notes 103, 724–736 (2018). https://doi.org/10.1134/S000143461805005X
D. T. Luyen and N. M. Tri, ‘‘Existence of infinitely many solutions for semilinear degenerate Schrödinger equations,’’ J. Math. Anal. Appl. 461, 1271–1286 (2018). https://doi.org/10.1016/j.jmaa.2018.01.016
A. Mao and Z. Zhang, ‘‘Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition,’’ Nonlinear Anal.: Theory, Methods Appl. 70, 1275–1287 (2009). https://doi.org/10.1016/j.na.2008.02.011
P. H. Rabinowitz, ‘‘On a class of nonlinear Schrödinger equations,’’ Z. Angew. Math. Phys. 43, 270–291 (1992). https://doi.org/10.1007/BF00946631
P. H. Rabinowitz, Minimax Methods in Critical Point Theory With Applications to Differential Equations Regional Conference Series in Mathematics, Vol. 65, (American Mathematical Soc., Providence, R.I., 1986).
B. Rahal and M. K. Hamdani, ‘‘Infinitely many solutions for \(\Delta_{\alpha}\)-Laplace equations with sign-changing potential,’’ J. Fixed Point Theory Appl. 20, 137 (2018). https://doi.org/10.1007/s11784-018-0617-3
X. H. Tang, ‘‘Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity,’’ J. Math. Anal. Appl. 401, 407–415 (2013). https://doi.org/10.1016/j.jmaa.2012.12.035
J. Zhang, X. Tang, and W. Zhang, ‘‘Existence of multiple solutions of Kirchhoff type equation with sign-changing potential,’’ Appl. Math. Comput. 242, 491–499 (2014). https://doi.org/10.1016/j.amc.2014.05.070
Funding
This work is supported by Research Fund of National Natural Science Foundation of China (no. 11861046), Chongqing Municipal Education Commission (no. KJQN20190081), Chongqing Technology and Business University(no. CTBUZDPTTD201909), Graduate Innovation Project of Chongqing Technology and Business University (yjscxx2021-112-109).
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Chen, J., Li, L. & Chen, S. Infinitely Many Solutions for Kirchhoff-Type Equations Involving Degenerate Operator. J. Contemp. Mathemat. Anal. 57, 252–266 (2022). https://doi.org/10.3103/S1068362322040045
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DOI: https://doi.org/10.3103/S1068362322040045