Existence of solutions for a class of Kirchhoff-type equations with indefinite potential
Authors
-
linlian Xiao
School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550025, China
https://orcid.org/0009-0004-4200-9115
-
Jiaqian Yuan
School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550025, China
Corresponding Author
https://orcid.org/0009-0006-5494-4367
-
Jian Zhou
School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550025, China
Corresponding Author
https://orcid.org/0000-0002-8013-2622
-
Yunshun Wu
School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550025, China
Corresponding Author
https://orcid.org/0000-0001-7587-2717
DOI:
https://doi.org/10.61383/ejam.20242368Keywords:
Kirchhoff-type equations, (C)_c-condition, Symmetric Mountain Pass TheoremAbstract
This study explores the existence of solutions to the following Kirchhoff-type problem
\[
\left\{
\begin{array}
[c]{ll}%
-\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+ V(x)u=f(x,u),~{\rm{in}}~ \mathbb{R}^{3},\\
u\in H^1(\mathbb{R}^3),%
\end{array} %
\right.
\]
where a and b are postive constants, and the potential \(V(x)\) is continuous and indefinite in sign. With suitable assumptions on
\(V(x)\) and \(f\), we establish the existence of solutions using the Symmetric Mountain Pass Theorem.
References
G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
Lions J L, On some questions in boundary value problems of mathematical physics, North-Holland Math. Studies. North-Holland, 1978.
Bernstein S, Sur une classe d’equations fonctionnelles aux d ´ eriv ´ ees partielles, Izv. Ross. Akad. Nauk. Seriya ´ Mat. 4(1)(1940), no. 1, 17-26.
Arosio A, Panizzi S, On the well-posedness of the Kirchhoff string, Trans. Am. Math. Soc. 48(1)(1996), no. 1, 305-330, DOI: 10.1090/s0002-9947-96-01532-2.
Cavalcanti M M, Domingos Cavalcanti V N, Soriano J A, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, (2001), DOI: 10.57262/ade/1357140586.
X Wu, Existence of nontrivial solutions and high energy solutions for Schrodinger–Kirchhoff-type equations in ¨ RN, Nonlinear Anal.: Real World Appl. 12(2)(2011), no. 2, 1278-1287, DOI: 10.1016/j.nonrwa.2010.09. 023.
C Chen, Y Kuo, T Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differ. Equ. 250(4)(2011), no. 4, 1876-1908, DOI: 10.1016/j.jde.2010.11.017.
Alves C O, Figueiredo GM, Nonlinear perturbations of a periodic Kirchhoff equation in RN, Nonlinear Anal.: Theory, Methods Appl. 75(5)(2012), 2750-2759, DOI: 10.1016/j.na.2011.11.017.
Y Li, F Li, J Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equ. 253(7)(2012), no. 7, 2285-2294, DOI: 10.1016/j.jde.2012.05.017.
G Li, H Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3, J. Differ. Equ. 257(2)(2014), no.2, 566-600, DOI: 10.1016/j.jde.2014.04.011.
Q Li, X Wu, A new result on high energy solutions for Schrodinger–Kirchhoff type equations in ¨ RN, Appl. Math. Lett. 30(2014), 24-27, DOI: 10.1016/j.aml.2013.12.002.
B Cheng, A New Result on Multiplicity of Nontrivial Solutions for the Nonhomogenous Schrodinger–Kirchhoff ¨ Type Problem in RN, Mediterr. J. Math. 13(2016), 1099-1116, DOI: 10.1007/s00009-015-0527-1.
C Ji, F Fang, Zhang, B.: A multiplicity result for asymptotically linear Kirchhoff equations, Adv. Nonlinear Anal. 8(1)(2017), no.1, 267-277, DOI: 10.1515/anona-2016-0240.
Batista A M, Furtado M F, Existence of solution for an asymptotically linear Schrodinger-Kirchhoff equation, ¨ Potential Anal. 50(2019), 609-619, DOI: 10.1007/s11118-018-9697-3.
Y Zhang, X Tang, D Qin, Infinitely many solutions for Kirchhoff problems with lack of compactness, Nonlinear Anal. 197(2020), 111856, DOI: 10.1016/j.na.2020.111856.
W Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differ. Equ.259(4)(2015), 1256-1274, DOI: 10.1016/j.jde.2015.02.040.
S Chen, S Liu, Standing waves for 4-superlinear Schrodinger-Kirchhoff equations, Math. Methods Appl. Sci. ¨38(11)(2015), no. 11, 2185-2193, DOI: 10.1002/mma.3212.
Y Wu, S Liu, Existence and multiplicity of solutions for asymptotically linear Schrodinger–Kirchhoff equations, ¨Nonlinear Anal.: Real World Appl. 26(2015), 191-198, DOI: 10.1016/j.nonrwa.2015.05.010.
S Chen, X Tang, Infinitely many solutions for super-quadratic Kirchhoff-type equations with sign-changing potential, Appl. Math. Lett. 67(2017), 40-45, DOI: 10.1016/j.aml.2016.12.003.
J Sun, L Li, Cencelj M, et al, Infinitely many sign-changing solutions for Kirchhoff type problems in R3, Nonlinear Anal. 186(2019), 33-54, DOI: 10.1016/j.na.2018.10.007.
J Zhou, Y Wu, Existence of solutions for a class of Kirchhoff-type equations with indefinite potential, Bound. Value Probl. 2021(1)(2021), no. 1, 1-13, DOI: 10.1186/s13661-021-01550-5.
S Jiang, S Liu, Multiple solutions for Schrodinger–Kirchhoff equations with indefinite potential, Appl. Math. ¨ Lett. 124(2022), 107672, DOI: 10.1016/j.aml.2021.107672.
S Jiang, S Liu, Infinitely many solutions for indefinite Kirchhoff equations and Schrodinger–Poisson systems, ¨ Appl. Math. Lett. 141(2023), 108620, DOI: 10.1016/j.aml.2023.108620.
W Chen, Y Wu, Jhang S, On nontrivial solutions of nonlinear Schrodinger equations with sign-changing ¨ potential, Adv. Differ. Equ. 2021(1)(2021), 232, DOI: 10.1186/s13662-021-03390-0.
J Sun, T Wu, On the nonlinear Schrodinger–Poisson systems with sign-changing potential, Z. f ¨ ur angew. Math. ¨ und Phys. 66(2015), 1649-1669, DOI: 10.1007/s00033-015-0494-1.
Rabinowitz P H, Minimax methods in critical point theory with applications to differential equations, Am. Math. Soc. 1986.
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Copyright (c) 2024 linlian Xiao; Jiaqian Yuan, Jian Zhou, Yunshun Wu (Corresponding Author)
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