Sign-changing solutions of critical quasilinear Kirchhoff-Schrödinger-Poisson system with logarithmic nonlinearity

Hui Jian; Shenghao Feng; Li Wang; Hui Jian; Shenghao Feng; Li Wang

doi:10.3934/math.2023431

AIMS Mathematics

2023, Volume 8, Issue 4: 8580-8609. doi: 10.3934/math.2023431

Previous Article Next Article

Research article Special Issues

Sign-changing solutions of critical quasilinear Kirchhoff-Schrödinger-Poisson system with logarithmic nonlinearity

1.
College of Science, East China Jiaotong University, Nanchang 330013, Jiangxi, China
2.
Department of Mathematics, Nanchang University, Nanchang 330031, Jiangxi, China

Received: 25 November 2022 Revised: 16 January 2023 Accepted: 27 January 2023 Published: 06 February 2023
MSC : 35A15, 35J60, 47G20

In the present paper, we study the following Kirchhoff-Schrödinger-Poisson system with logarithmic and critical nonlinearity:

$ \begin{align} \begin{array}{ll} \left \{ \begin{array}{ll} - \Bigr(a+b\int_\Omega|\nabla u|^2{\mathrm{d}}x \Bigr)\Delta u+V(x)u-\frac{1}{2}u\Delta (u^2)+\phi u = \lambda |u|^{q-2}u\ln|u|^2+|u|^4u, &x\in \Omega, \\ -\Delta \phi = u^2,& x\in \Omega, \\ u = \phi = 0,& x\in \partial\Omega, \end{array} \right . \end{array} \end{align} $

where $ \lambda, b > 0, a > \frac{1}{4}, 4 < q < 6, $ $ V(x) $ is a smooth potential function and $ \Omega $ is a bounded domain in $ \mathbb{R}^3 $ with Lipschitz boundary. Combining constraint variational method and perturbation method, we prove that the above problem has a least energy sign-changing solution $ u_0 $ which has precisely two nodal domains. Moreover, we show that the energy of $ u_0 $ is strictly larger than twice the ground state energy.
- quasilinear Kirchhoff-Schrödinger-Poisson,
- critical problem,
- logarithmic nonlinearity
Citation: Hui Jian, Shenghao Feng, Li Wang. Sign-changing solutions of critical quasilinear Kirchhoff-Schrödinger-Poisson system with logarithmic nonlinearity[J]. AIMS Mathematics, 2023, 8(4): 8580-8609. doi: 10.3934/math.2023431

Related Papers:

Abstract

In the present paper, we study the following Kirchhoff-Schrödinger-Poisson system with logarithmic and critical nonlinearity:

$ \begin{align} \begin{array}{ll} \left \{ \begin{array}{ll} - \Bigr(a+b\int_\Omega|\nabla u|^2{\mathrm{d}}x \Bigr)\Delta u+V(x)u-\frac{1}{2}u\Delta (u^2)+\phi u = \lambda |u|^{q-2}u\ln|u|^2+|u|^4u, &x\in \Omega, \\ -\Delta \phi = u^2,& x\in \Omega, \\ u = \phi = 0,& x\in \partial\Omega, \end{array} \right . \end{array} \end{align} $

where $ \lambda, b > 0, a > \frac{1}{4}, 4 < q < 6, $ $ V(x) $ is a smooth potential function and $ \Omega $ is a bounded domain in $ \mathbb{R}^3 $ with Lipschitz boundary. Combining constraint variational method and perturbation method, we prove that the above problem has a least energy sign-changing solution $ u_0 $ which has precisely two nodal domains. Moreover, we show that the energy of $ u_0 $ is strictly larger than twice the ground state energy.

References

[1]	C. O. Alves, F. J. S. A. Correa, On existence of solutions for a class of problem involving a nonlinear operator, Comm. Appl. Nonlinear Anal., 8 (2001), 43–56.
[2]	A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305–330. http://dx.doi.org/10.1090/S0002-9947-96-01532-2 doi: 10.1090/S0002-9947-96-01532-2
[3]	T. Bartsch, T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. I. H. Poincare Anal. NonLineaire, 22 (2005), 259–281. http://dx.doi.org/10.1016/j.anihpc.2004.07.005 doi: 10.1016/j.anihpc.2004.07.005
[4]	M. M. Cavalcanti, V. N. D. Cavalcanti, J. A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equ., 6 (2001), 701–730. http://dx.doi.org/10.57262/ade/1357140586 doi: 10.57262/ade/1357140586
[5]	J. Chen, X. Tang, Z. Gao, B. Chen, Ground state sign-changing solutions for a class of generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation, J. Fixed Point Theory A., 19 (2017), 3127–3149. http://dx.doi.org/10.1007/s11784-017-0475-4 doi: 10.1007/s11784-017-0475-4
[6]	P. D'Ancona, S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247–262. http://dx.doi.org/10.1007/BF02100605 doi: 10.1007/BF02100605
[7]	Y. Deng, W. Shuai, Sign-changing solutions for non-local elliptic equations involving the fractional Laplacian, Adv. Differential Equ., 23 (2018), 109–134. http://dx.doi.org/10.57262/ade/1508983363 doi: 10.57262/ade/1508983363
[8]	X. Feng, Y. Zhang, Existence of non-trivial solution for a class of modified Schrödinger-Poisson equations via perturbation method, J. Math. Anal. Appl., 442 (2016), 673–684. http://dx.doi.org/10.1016/j.jmaa.2016.05.002 doi: 10.1016/j.jmaa.2016.05.002
[9]	G. M. Figueiredo, G. Siciliano, Existence and asymptotic behaviour of solutions for a quasilinear Schrödinger-Poisson system under a critical nonlinearity, arXiv: 1707.05353, 2017. https://doi.org/10.48550/arXiv.1707.05353
[10]	G. M. Figueiredo, G. Siciliano, Quasilinear Schrödinger-Poisson system under an exponential critical nonlinearity: existence and asymptotic of solutions, Arch. Math., 112 (2019), 313–327. http://dx.doi.org/10.1007/s00013-018-1287-5 doi: 10.1007/s00013-018-1287-5
[11]	X. He, W. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat.Pura. Appl., 193 (2014), 473–500. http://dx.doi.org/10.1007/s10231-012-0286-6 doi: 10.1007/s10231-012-0286-6
[12]	R. Illner, O. Kavian, H. Lange, Stationary solutions of quasi-linear Schrödinger-Poisson systems, J. Differential Equ., 145 (1998), 1–16. http://dx.doi.org/10.1006/jdeq.1997.3405 doi: 10.1006/jdeq.1997.3405
[13]	G. Kirchhoff, Mechanik, Leipzig: Teubner, 1883.
[14]	Y. Li, D. Wang, J. Zhang, Sign-changing solutions for a class of $p$-Laplacian Kirchhoff-type problem with logarithmic nonlinearity, AIMS Math., 5 (2020), 2100–2112. http://dx.doi.org/10.3934/math.2020139 doi: 10.3934/math.2020139
[15]	F. Li, X. Zhu, Z. Liang, Multiple solutions to a class of generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation, J. Math. Anal. Appl., 443 (2016), 11–38. http://dx.doi.org/10.1016/j.jmaa.2016.05.005 doi: 10.1016/j.jmaa.2016.05.005
[16]	S. Liang, V. D. R$\breve{a}$dulescu, Least-energy nodal solutions of critical Kirchhoff problems with logarithmic nonlinearity, Anal. Math. Phys., 10 (2020), 45. http://dx.doi.org/10.1007/s13324-020-00386-z doi: 10.1007/s13324-020-00386-z
[17]	J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284–346.
[18]	X. Liu, J. Liu, Z. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equ., 254 (2013), 102–124. http://dx.doi.org/10.1016/j.jde.2012.09.006 doi: 10.1016/j.jde.2012.09.006
[19]	M. Massar, On a nonlocal Schrödinger-Poisson system with critical exponent, Appl. Math. E-Notes, 21 (2021), 44–52.
[20]	C. Miranda, Unosservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5–7.
[21]	A. Nakamura, Damping and modification of exciton solitary waves, J. Phys. Soc. Japan, 42 (1977), 1824–1835. http://doi.org/10.1143/JPSJ.42.1824 doi: 10.1143/JPSJ.42.1824
[22]	W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equ., 259 (2015), 1256–1274. http://dx.doi.org/10.1016/j.jde.2015.02.040 doi: 10.1016/j.jde.2015.02.040
[23]	K. Susumu, Large-Amplitude Quasi-Solitons in Superfluid Films, J. Phys. Soc. Japan, 50 (1981), 3262–3267. http://doi.org/10.1143/JPSJ.50.3262 doi: 10.1143/JPSJ.50.3262
[24]	X. Tang, B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equ., 261 (2016), 2384–2402. http://dx.doi.org/10.1016/j.jde.2016.04.032 doi: 10.1016/j.jde.2016.04.032
[25]	K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equ., 261 (2016), 3061–3106. http://dx.doi.org/10.1016/j.jde.2016.05.022 doi: 10.1016/j.jde.2016.05.022
[26]	D. Wang, Least energy sign-changing solutions of Kirchhoff-type equation with critical growth, J. Math. Phys, 61 (2020), 011501. http://dx.doi.org/10.1063/1.5074163 doi: 10.1063/1.5074163
[27]	L. Wang, On a quasilinear Schrödinger-Kirchhoff-type equation with radial potentials, Nonlinear Anal., 83 (2013), 58–68. http://dx.doi.org/10.1016/j.na.2012.12.012 doi: 10.1016/j.na.2012.12.012
[28]	M. Willem, Minimax theorems, In: Progress in Nonlinear Differential Equations and their Applications, Boston: Birkh$\ddot{a}$user, 24 (1996).

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)