Ground states of a Kirchhoff equation with the potential on the lattice graphs

Wenqian Lv; Wenqian Lv

doi:10.3934/cam.2023038

Communications in Analysis and Mechanics

2023, Volume 15, Issue 4: 792-810. doi: 10.3934/cam.2023038

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Research article

Ground states of a Kirchhoff equation with the potential on the lattice graphs

Wenqian Lv ^,

School of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

Received: 25 August 2023 Revised: 22 October 2023 Accepted: 31 October 2023 Published: 14 November 2023
35J60, 35J20, 35R02

In this paper, we study the nonlinear Kirchhoff equation

$ \begin{align*} -\Big(a+b\int_{\mathbb{Z}^{3}}|\nabla u|^{2} d \mu\Big)\Delta u+V(x)u = f(u) \end{align*} $

on lattice graph $ \mathbb{Z}^3 $, where $ a, b > 0 $ are constants and $ V:\mathbb{Z}^{3}\rightarrow \mathbb{R} $ is a positive function. Under a Nehari-type condition and 4-superlinearity condition on $ f $, we use the Nehari method to prove the existence of ground-state solutions to the above equation when $ V $ is coercive. Moreover, we extend the result to noncompact cases in which $ V $ is a periodic function or a bounded potential well.
- Kirchhoff equation,
- lattice graph,
- ground states,
- Nehari manifold,
- variational methods
Citation: Wenqian Lv. Ground states of a Kirchhoff equation with the potential on the lattice graphs[J]. Communications in Analysis and Mechanics, 2023, 15(4): 792-810. doi: 10.3934/cam.2023038

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Abstract

In this paper, we study the nonlinear Kirchhoff equation

$ \begin{align*} -\Big(a+b\int_{\mathbb{Z}^{3}}|\nabla u|^{2} d \mu\Big)\Delta u+V(x)u = f(u) \end{align*} $

on lattice graph $ \mathbb{Z}^3 $, where $ a, b > 0 $ are constants and $ V:\mathbb{Z}^{3}\rightarrow \mathbb{R} $ is a positive function. Under a Nehari-type condition and 4-superlinearity condition on $ f $, we use the Nehari method to prove the existence of ground-state solutions to the above equation when $ V $ is coercive. Moreover, we extend the result to noncompact cases in which $ V $ is a periodic function or a bounded potential well.

References

[1]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 2019,267–277.
[2]	J. Lions, On some questions in boundary value problems of mathmatical phisics, North-Holland Math. Stud., 30 (1978), 284–346. https://doi.org/10.1016/S0304-0208(08)70870-3 doi: 10.1016/S0304-0208(08)70870-3
[3]	A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349–381. https://doi.org/10.1016/0022-1236(73)90051-7 doi: 10.1016/0022-1236(73)90051-7
[4]	J. Jin, X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbb{R}^N$, J. Math. Anal. Appl., 369 (2010), 564–574. https://doi.org/10.1016/j.jmaa.2010.03.059 doi: 10.1016/j.jmaa.2010.03.059
[5]	P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS, Reg. Conf. Ser. Math. 65, Amer. Math. Soc., Providence, RI, 1986. https://doi.org/10.1090/cbms/065
[6]	X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbb{R}^N$, Nonlinear Anal. Real World Appl., 12 (2011), 1278–1287. https://doi.org/10.1016/j.nonrwa.2010.09.023 doi: 10.1016/j.nonrwa.2010.09.023
[7]	X. He, W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$, J. Differ. Equ., 252 (2012), 1813–1834. https://doi.org/10.1016/j.jde.2011.08.035 doi: 10.1016/j.jde.2011.08.035
[8]	A. Mao, Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275–1287. https://doi.org/10.1016/j.na.2008.02.011 doi: 10.1016/j.na.2008.02.011
[9]	A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305–330. https://doi.org/10.1090/S0002-9947-96-01532-2 doi: 10.1090/S0002-9947-96-01532-2
[10]	G. Che, T. Wu, Multiple positive solutions for a class of Kirchhoff type equations with indefinite nonlinearities, Adv. Nonlinear Anal., 11 (2022), 598–619. https://doi.org/10.1515/anona-2021-0213 doi: 10.1515/anona-2021-0213
[11]	H. Guo, Y. Zhang, H. S. Zhou, Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential, Commun. Pure Appl. Anal. 17 (2018), 1875–1897. https://doi.org/10.3934/cpaa.2018089 doi: 10.3934/cpaa.2018089
[12]	X. He, W. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl., 193 (2014), 473–500. https://doi.org/10.1007/s10231-012-0286-6 doi: 10.1007/s10231-012-0286-6
[13]	C. Ji, F. Fang, B. Zhang, A multiplicity result for asymptotically linear Kirchhoff equations, Adv. Nonlinear Anal., 8 (2019), 267–277. https://doi.org/10.1515/anona-2016-0240 doi: 10.1515/anona-2016-0240
[14]	G. Li, H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^3$, J. Differ. Equ., 257 (2014), 566–600. https://doi.org/10.1016/j.jde.2014.04.011 doi: 10.1016/j.jde.2014.04.011
[15]	Y. Li, F. Li, J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equ., 253 (2012), 2285–2294. https://doi.org/10.1016/j.jde.2012.05.017 doi: 10.1016/j.jde.2012.05.017
[16]	X. Ma, X. He, Nontrivial solutions for Kirchhoff equations with periodic potentials, Electron. J. Differential Equations, 102 (2016), 2 67–277.
[17]	K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ., 221 (2006), 246–255. https://doi.org/10.1016/j.jde.2005.03.006 doi: 10.1016/j.jde.2005.03.006
[18]	X. Tang, B. Chen, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differ. Equ., 261 (2016), 2384–2402. https://doi.org/10.1016/j.jde.2016.04.032 doi: 10.1016/j.jde.2016.04.032
[19]	A. Grigor'yan, Y. Lin, Y. Yang, Yamabe type equations on graphs, J. Differ. Equ., 261 (2016), 4924–4943. https://doi.org/10.1016/j.jde.2016.07.011 doi: 10.1016/j.jde.2016.07.011
[20]	A. Grigor'yan, Y. Lin, Y. Yang, Existence of positive solutions to some nonlinear equations on locally finite graphs, Sci. China Math., 60 (2017), 1311–1324. https://doi.org/10.1007/s11425-016-0422-y doi: 10.1007/s11425-016-0422-y
[21]	N. Zhang, L. Zhao, Convergence of ground state solutions for nonlinear Schrödinger equations on graphs, Sci. China Math., 61 (2018), 1481–1494. https://doi.org/10.1007/s11425-017-9254-7 doi: 10.1007/s11425-017-9254-7
[22]	X. Han, M. Shao, L. Zhao, Existence and convergence of solutions for nonlinear biharmonic equations on graphs, J. Differ. Equ., 268 (2020), 3936–3961. https://doi.org/10.1016/j.jde.2019.10.007 doi: 10.1016/j.jde.2019.10.007
[23]	B. Hua, R. Li, The existence of extremal functions for discrete Sobolev inequalities on lattice graphs, J. Differ. Equ., 305 (2021), 224–241. https://doi.org/10.1016/j.jde.2021.10.016 doi: 10.1016/j.jde.2021.10.016
[24]	B. Hua, D. Mugnolo, Time regularity and long-time behavior of parabolic p-Laplace equations on infinite graphs, J. Differ. Equ., 259 (2015), 6162–6190. https://doi.org/10.1016/j.jde.2015.07.018 doi: 10.1016/j.jde.2015.07.018
[25]	B. Hua, W Xu, Existence of ground state solutions to some Nonlinear Schrödinger equations on lattice graphs, Calc. Var., 62 (2023), 127. https://doi.org/10.1007/s00526-023-02470-1 doi: 10.1007/s00526-023-02470-1
[26]	Y. Li, Z. Wang, J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 829–837. https://doi.org/10.1016/j.anihpc.2006.01.003 doi: 10.1016/j.anihpc.2006.01.003
[27]	A. Szulkin, T. Weth, The method of Nehari manifold, in Analysis and Applications, International Press, (2020), 2314–2351.
[28]	J. Zhang, W. Zhang, Semiclassical states for coupled nonlinear Schrodinger system with competing potentials, J. Geom. Anal., 32 (2022), 114. https://doi.org/10.1007/s12220-022-00870-x doi: 10.1007/s12220-022-00870-x
[29]	A. Szulkin, T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802–3822. https://doi.org/10.1016/j.jfa.2009.09.013 doi: 10.1016/j.jfa.2009.09.013
[30]	P. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984) 109–145. https://doi.org/10.1016/S0294-1449(16)30428-0 doi: 10.1016/S0294-1449(16)30428-0
[31]	P. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223–283. https://doi.org/10.1016/S0294-1449(16)30422-X doi: 10.1016/S0294-1449(16)30422-X
[32]	A. Grigor'yan, Y. Lin, Y. Yang, Kazdan-Warner equation on graph, Calc. Var., 55 (2016), 92. https://doi.org/10.1007/s00526-016-1042-3 doi: 10.1007/s00526-016-1042-3
[33]	M. Ostrovskii, Sobolev spaces on graphs, Quaest. Math., 28 (2005), 501–523. https://doi.org/10.2989/16073600509486144 doi: 10.2989/16073600509486144
[34]	B. Hua, R. Li, L. Wang, A class of semilinear elliptic equations on lattice graphs, J. Differ. Equ., 363 (2022), 327–349. https://doi.org/10.48550/arXiv.2203.05146 doi: 10.48550/arXiv.2203.05146
[35]	M. Willen, Minimax Theorems, Birkhäuser, Boston, 1996. https://doi.org/10.1007/978-1-4612-4146-1

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