Least energy sign-changing solutions for Kirchhoff-Schrödinger-Poisson system on bounded domains

Xia Su; Wen Guan; Xia Li; Xia Su; Wen Guan; Xia Li

doi:10.3934/era.2023149

Electronic Research Archive

2023, Volume 31, Issue 5: 2959-2973. doi: 10.3934/era.2023149

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Least energy sign-changing solutions for Kirchhoff-Schrödinger-Poisson system on bounded domains

Xia Su ^{1
,
,},
Wen Guan ²,
Xia Li ²

1.
Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huai'an 223003, Jiangsu, China
2.
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, Gansu, China

Received: 28 October 2022 Revised: 16 February 2023 Accepted: 20 February 2023 Published: 20 March 2023

We investigate the following nonlinear system

$ \begin{cases} -(a+ b\int _{\Omega}|\nabla u|^{2}dx)\Delta u+\phi u = \lambda u+\mu|u|^{2}u, \; \ x\in\Omega, \\ -\Delta\phi = u^{2}, \; \ x\in\Omega, \\ u = \phi = 0, \; \ x\in \partial\Omega, \end{cases} $

with $ a, b > 0 $, $ \lambda, \mu\in\mathbb{R} $, and $ \Omega\subset \mathbb{R}^{3} $ is bounded with smooth boundary. Let $ \lambda_{1} > 0 $ be the first eigenvalue of $ (-\Delta u, H^{1}_{0}(\Omega)) $. We get that for certain $ \widetilde{\mu} > 0 $ there exists at least one least energy sign-changing solution for the above system if $ \lambda < a\lambda_{1} $ and $ \mu > \widetilde{\mu} $. In addition, we remark that the nonlinearity $ \lambda u+\mu|u|^{2}u $ does not satisfy the growth conditions.
- Kirchhoff type equation,
- Schrödinger-Poisson problem,
- sign-changing solutions,
- Nehari manifold,
- the growth conditions
Citation: Xia Su, Wen Guan, Xia Li. Least energy sign-changing solutions for Kirchhoff-Schrödinger-Poisson system on bounded domains[J]. Electronic Research Archive, 2023, 31(5): 2959-2973. doi: 10.3934/era.2023149

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Abstract

We investigate the following nonlinear system

$ \begin{cases} -(a+ b\int _{\Omega}|\nabla u|^{2}dx)\Delta u+\phi u = \lambda u+\mu|u|^{2}u, \; \ x\in\Omega, \\ -\Delta\phi = u^{2}, \; \ x\in\Omega, \\ u = \phi = 0, \; \ x\in \partial\Omega, \end{cases} $

with $ a, b > 0 $, $ \lambda, \mu\in\mathbb{R} $, and $ \Omega\subset \mathbb{R}^{3} $ is bounded with smooth boundary. Let $ \lambda_{1} > 0 $ be the first eigenvalue of $ (-\Delta u, H^{1}_{0}(\Omega)) $. We get that for certain $ \widetilde{\mu} > 0 $ there exists at least one least energy sign-changing solution for the above system if $ \lambda < a\lambda_{1} $ and $ \mu > \widetilde{\mu} $. In addition, we remark that the nonlinearity $ \lambda u+\mu|u|^{2}u $ does not satisfy the growth conditions.

References

[1]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[2]	D. Cassani, Z. Liu, C. Tarsi, J. Zhang, Multiplicity of sign-changing solutions for Kirchhoff-type equations, Nonlinear Anal., 186 (2019), 145–161. https://doi.org/10.1016/j.na.2019.01.025 doi: 10.1016/j.na.2019.01.025
[3]	B. Cheng, X. H. Tang, Ground state sign-changing solutions for asymptotically 3-linear Kirchhoff-type problems, Complex Var. Elliptic Equations, 62 (2017), 1093–1116. https://doi.org/10.1080/17476933.2016.1270272 doi: 10.1080/17476933.2016.1270272
[4]	Y. B. Deng, S. J. Peng, W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}^{3}$, J. Funct. Anal., 269 (2015), 3500–3527. https://doi.org/10.1016/j.jfa.2015.09.012 doi: 10.1016/j.jfa.2015.09.012
[5]	G. M. Figueiredo, J. R. Santos Júnior, Existence of a least energy nodal solution for a Schrödinger-Kirchhoff equation with potential vanishing at infinity, J. Math. Phys., 56 (2015), 051506. https://doi.org/10.1063/1.4921639 doi: 10.1063/1.4921639
[6]	F. Y. Li, C. Gao, X. Zhu, Existence and concentration of sign-changing solutions to Kirchhoff-type system with Hartree-type nonlinearity, J. Math. Anal. Appl., 448 (2017), 60–80. https://doi.org/10.1016/j.jmaa.2016.10.069 doi: 10.1016/j.jmaa.2016.10.069
[7]	Q. Li, X. Du, Z. Zhao, Existence of sign-changing solutions for nonlocal Kirchhoff-Schrödinger-type equations in $\mathbb{R}^{3}$, J. Math. Anal. Appl., 477 (2019), 174–186. https://doi.org/10.1016/j.jmaa.2019.04.025 doi: 10.1016/j.jmaa.2019.04.025
[8]	S. Lu, Signed and sign-changing solutions for a Kirchhoff-type equation in bounded domains, J. Math. Anal. Appl., 432 (2015), 965–982. https://doi.org/10.1016/j.jmaa.2015.07.033 doi: 10.1016/j.jmaa.2015.07.033
[9]	A. M. Mao, S. X. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239–243. https://doi.org/10.1016/j.jmaa.2011.05.021 doi: 10.1016/j.jmaa.2011.05.021
[10]	A. M. Mao, Z. T. 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
[11]	M. Shao, A. Mao, Signed and sign-changing solutions of Kirchhoff type problems, J. Fixed. Point Theory Appl., 20 (2018), 1–20. https://doi.org/10.1007/s11784-018-0486-9 doi: 10.1007/s11784-018-0486-9
[12]	W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differ. Equations, 259 (2015), 1256–1274. https://doi.org/10.1016/j.jde.2015.02.040 doi: 10.1016/j.jde.2015.02.040
[13]	J. Sun, L. Li, M. Cencelj, B. Gabrovšek, Infinitely many sign-changing solutions for Kirchhoff type problems in $\mathbb{R}^{3}$, Nonlinear Anal., 186 (2019), 33–54. https://doi.org/10.1016/j.na.2018.10.007 doi: 10.1016/j.na.2018.10.007
[14]	X. H. Tang, B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differ. Equations, 261 (2016), 2384–2402. https://doi.org/10.1016/j.jde.2016.04.032 doi: 10.1016/j.jde.2016.04.032
[15]	D. B. Wang, Least energy sign-changing solutions of Kirchhoff-type equation with critical growth, J. Math. Phys., 61 (2020), 011501. https://doi.org/10.1063/1.5074163 doi: 10.1063/1.5074163
[16]	L. Wang, B. L. Zhang, K. Cheng, Ground state sign-changing solutions for the Schrödinger-Kirchhoff equation in $\mathbb{R}^{3}$, J. Math. Anal. Appl., 466 (2018), 1545–1569. https://doi.org/10.1016/j.jmaa.2018.06.071 doi: 10.1016/j.jmaa.2018.06.071
[17]	H. Ye, The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 431 (2015), 935–954. https://doi.org/10.1016/j.jmaa.2015.06.012 doi: 10.1016/j.jmaa.2015.06.012
[18]	Z. T. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descentow, J. Math. Anal. Appl., 317 (2006), 456–463. https://doi.org/10.1016/j.jmaa.2005.06.102 doi: 10.1016/j.jmaa.2005.06.102
[19]	X. Zhong, C. L. Tang, The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem, Commun. Pure Appl. Anal., 16 (2017), 611–628. https://doi.org/10.3934/cpaa.2017030 doi: 10.3934/cpaa.2017030
[20]	C. O. Alves, M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153–1166. https://doi.org/10.1007/s00033-013-0376-3 doi: 10.1007/s00033-013-0376-3
[21]	S. Chen, X. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb{R}^{3}$, Z. Angew. Math. Phys., 67 (2016). https://doi.org/10.1007/s00033-016-0695-2 doi: 10.1007/s00033-016-0695-2
[22]	S. Khoutir, Least energy sign-changing solutions for a class of Schrödinger-Poisson system on bounded domains, J. Math. Phys., 62 (2021), 031509. https://doi.org/10.1063/5.0040741 doi: 10.1063/5.0040741
[23]	S. Kim, J. Seok, On nodal solutions of the nonlinear Schrödinger-Poisson equations, Commun. Contemp. Math., 14 (2012), 1250041. https://doi.org/10.1142/S0219199712500411 doi: 10.1142/S0219199712500411
[24]	Z. Liang, J. Xu, X. Zhu, Revisit to sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^{3}$, J. Math. Anal. Appl., 435 (2016), 783–799. https://doi.org/10.1016/j.jmaa.2015.10.076 doi: 10.1016/j.jmaa.2015.10.076
[25]	Z. Liu, Z. Wang, J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 4 (2016), 775–794. https://doi.org/10.1007/s10231-015-0489-8 doi: 10.1007/s10231-015-0489-8
[26]	W. Shuai, Q. Wang, Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^{3}$, Z. Angew. Math. Phys., 66 (2015), 3267–3282. https://doi.org/10.1007/s00033-015-0571-5 doi: 10.1007/s00033-015-0571-5
[27]	D. B. Wang, H. Zhang, W. Guan, Existence of least-energy sign-changing solutions for Schrödinger-Poisson system with critical growth, J. Math. Anal. Appl., 479 (2019), 2284–2301. https://doi.org/10.1016/j.jmaa.2019.07.052 doi: 10.1016/j.jmaa.2019.07.052
[28]	Z. Wang, H. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^{3}$, Calc. Var. Partial Differ. Equations, 52 (2015), 927–943. https://doi.org/10.1007/s00526-014-0738-5 doi: 10.1007/s00526-014-0738-5
[29]	X. Zhong, C. L. Tang, Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in $\mathbb{R}^{3}$, Nonlinear Anal. Real World Appl., 39 (2018), 166–184. https://doi.org/10.1016/j.nonrwa.2017.06.014 doi: 10.1016/j.nonrwa.2017.06.014
[30]	V. Bobkov, Least energy nodal solutions for elliptic equations with indefinite nonlinearity, Electron. J. Qual. Theory Differ. Equations, 2014 (2014), 1–15. https://doi.org/10.14232/ejqtde.2014.1.56 doi: 10.14232/ejqtde.2014.1.56
[31]	D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655–674. https://doi.org/10.1016/j.jfa.2006.04.005 doi: 10.1016/j.jfa.2006.04.005
[32]	M. Willem, Minimax Theorems, Birkhäuser, Bosten, 1996.

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