Sign-changing solutions for the Schrödinger-Poisson system with concave-convex nonlinearities in $ \mathbb{R}^{3} $

Chen Yang; Chun-Lei Tang; Chen Yang; Chun-Lei Tang

doi:10.3934/cam.2023032

Communications in Analysis and Mechanics

2023, Volume 15, Issue 4: 638-657. doi: 10.3934/cam.2023032

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Research article

Sign-changing solutions for the Schrödinger-Poisson system with concave-convex nonlinearities in $ \mathbb{R}^{3} $

Chen Yang ,
Chun-Lei Tang ^,

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

Received: 14 September 2023 Revised: 10 October 2023 Accepted: 12 October 2023 Published: 19 October 2023
35A15, 35J20, 35J50

In this paper, we consider the following Schrödinger-Poisson system

$ \begin{equation*} \qquad \left\{ \begin{array}{ll} -\Delta u+V(x)u+\phi u = |u|^{p-2}u+ \lambda K(x)|u|^{q-2}u\ \ \ &\ \rm in\; \mathbb{R}^{3}, \\ -\Delta \phi = u^2 \ \ \ &\ \rm in\; \mathbb{R}^{3}.\ \end{array} \right. \end{equation*} $

Under the weakly coercive assumption on $ V $ and an appropriate condition on $ K $, we investigate the cases when the nonlinearities are of concave-convex type, that is, $ 1 < q < 2 $ and $ 4 < p < 6 $. By constructing a nonempty closed subset of the sign-changing Nehari manifold, we establish the existence of least energy sign-changing solutions provided that $ \lambda\in(-\infty, \lambda_*) $, where $ \lambda_* > 0 $ is a constant.
- Schrödinger-Poisson system,
- sign-changing solutions,
- concave-convex nonlinearities,
- variational method
Citation: Chen Yang, Chun-Lei Tang. Sign-changing solutions for the Schrödinger-Poisson system with concave-convex nonlinearities in $ \mathbb{R}^{3} $[J]. Communications in Analysis and Mechanics, 2023, 15(4): 638-657. doi: 10.3934/cam.2023032

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Abstract

In this paper, we consider the following Schrödinger-Poisson system

$ \begin{equation*} \qquad \left\{ \begin{array}{ll} -\Delta u+V(x)u+\phi u = |u|^{p-2}u+ \lambda K(x)|u|^{q-2}u\ \ \ &\ \rm in\; \mathbb{R}^{3}, \\ -\Delta \phi = u^2 \ \ \ &\ \rm in\; \mathbb{R}^{3}.\ \end{array} \right. \end{equation*} $

Under the weakly coercive assumption on $ V $ and an appropriate condition on $ K $, we investigate the cases when the nonlinearities are of concave-convex type, that is, $ 1 < q < 2 $ and $ 4 < p < 6 $. By constructing a nonempty closed subset of the sign-changing Nehari manifold, we establish the existence of least energy sign-changing solutions provided that $ \lambda\in(-\infty, \lambda_*) $, where $ \lambda_* > 0 $ is a constant.

References

[1]	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
[2]	A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519–543. https://doi.org/10.1006/jfan.1994.1078
[3]	A. Azzollini, A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90–108. https://doi.org/10.1016/j.jmaa.2008.03.057
[4]	T. Bartsch, A. Pankov, Z. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549–569. https://doi.org/10.1142/S0219199701000494
[5]	T. Bartsch, T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259–281. https://doi.org/10.1016/j.anihpc.2004.07.005 doi: 10.1016/j.anihpc.2004.07.005
[6]	V. Benci, D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283–293. http://dx.doi.org/10.12775/TMNA.1998.019
[7]	V. E. Bobkov, On existence of nodal solution to elliptic equations with convex-concave nonlinearities, Ufa Math. J., 5 (2013), 18–30. http://dx.doi.org/10.13108/2013-5-2-18
[8]	V. E. Bobkov, On the existence of a continuous branch of nodal solutions of elliptic equations with convex-concave nonlinearities, Differ. Equ., 50 (2014), 765–776. https://doi.org/10.1134/S0012266114060056
[9]	S. Chen, L. Li, Infinitely many large energy solutions for the Schrödinger-Poisson system with concave and convex nonlinearities, Appl. Math. Lett., 112 (2021), 106789. https://doi.org/10.1016/j.aml.2020.106789
[10]	S. Chen, X. Tang, Ground state sign-changing solutions for asymptotically cubic or super-cubic Schrödinger-Poisson systems without compact condition, Comput. Math. Appl., 74 (2017), 446–458. https://doi.org/10.1016/j.camwa.2017.04.031
[11]	T. D'Aprile, D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893–906. https://doi.org/10.1017/S030821050000353X doi: 10.1017/S030821050000353X
[12]	P. Drábek, S. I. Pohožaev, Positive solutions for the $p$-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703–726. https://doi.org/10.1017/S0308210500023787 doi: 10.1017/S0308210500023787
[13]	L. Gu, H. Jin, J. Zhang, Sign-changing solutions for nonlinear Schrödinger-Poisson systems with subquadratic or quadratic growth at infinity, Nonlinear Anal., 198 (2020), 111897. https://doi.org/10.1016/j.na.2020.111897
[14]	C. Lei, H. Suo, Positive solutions for a Schrödinger-Poisson system involving concave-convex nonlinearities, Comput. Math. Appl., 74 (2017), 1516–1524. https://doi.org/10.1016/j.camwa.2017.06.029
[15]	W. Li, V. D. Rădulescu, B. Zhang, Infinitely many solutions for fractional Kirchhoff-Schrödinger-Poisson systems, J. Math. Phys., 60 (2019), 011506. https://doi.org/10.1063/1.5019677
[16]	Z. Liu, L. Tao, D. Zhang, S. Liang, Y. Song, Critical nonlocal Schrödinger-Poisson system on the Heisenberg group, Adv. Nonlinear Anal., 11 (2022), 482–502. https://doi.org/10.1515/anona-2021-0203
[17]	Z. Liu, Z. Wang, Schrödinger equations with concave and convex nonlinearities, Z. Angew. Math. Phys., 56 (2005), 609–629. http://dx.doi.org/10.1007/s00033-005-3115-6
[18]	Z. Liu, Z. Wang, J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775–794. https://doi.org/10.1007/s10231-015-0489-8
[19]	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
[20]	Ó. Sánchez, J. Soler, Long-time dynamics of the Schröginger-Poisson-Salter system, J. Stat. Phys., 144 (2004), 179–204. https://doi.org/10.1023/B: JOSS.0000003109.97208.53
[21]	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
[22]	M. Shao, A. Mao, Multiplicity of solutions to Schrödinger-Poisson system with concave-convex nonlinearities, Appl. Math. Lett., 83 (2018), 212–218. https://doi.org/10.1016/j.aml.2018.04.005
[23]	M. Shao, A. Mao, Schrödinger-Poisson system with concave-convex nonlinearities, J. Math. Phys., 60 (2019), 061504. https://doi.org/10.1063/1.5087490
[24]	K. Sofiane, 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
[25]	J. Sun, T. Wu, Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger-Poisson system, J. Differential Equations, 260 (2016), 586–627. https://doi.org/10.1016/j.jde.2015.09.002 doi: 10.1016/j.jde.2015.09.002
[26]	J. Sun, T. Wu, On Schrödinger-Poisson systems involving concave-convex nonlinearities via a novel constraint approach, Commun. Contemp. Math., 23 (2021), 2050048. https://doi.org/10.1142/S0219199720500480
[27]	D. 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
[28]	X. Wang, F. Chen, F. Liao, Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential, Adv. Nonlinear Anal., 12, 20220319 (2023). https://doi.org/10.1515/anona-2022-0319
[29]	Z. Wang, H. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^{3}$, Calc. Var. Partial Differential Equations, 52 (2015), 927–943. https://doi.org/10.1007/s00526-014-0738-5 doi: 10.1007/s00526-014-0738-5
[30]	L. Wen, S. Chen, V. D. Rădulescu, Axially symmetric solutions of the Schrödinger-Poisson system with zero mass potential in $\mathbb R^2$, Appl. Math. Lett., 104, (2020), 106244. https://doi.org/10.1016/j.aml.2020.106244
[31]	T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differential Equations, 27 (2006), 421–437. https://doi.org/10.1007/s00526-006-0015-3 doi: 10.1007/s00526-006-0015-3
[32]	M. Willem, Minimax Theorems, Birkhäuser Boston, 1996. https://doi.org/10.1007/978-1-4612-4146-1
[33]	Z. Yang, Z. Ou, Nodal solutions for Schrödinger-Poisson systems with concave-convex nonlinearities, J. Math. Anal. Appl., 499 (2021), 125006. https://doi.org/10.1016/j.jmaa.2021.125006
[34]	S. Yu, Z. Zhang, Sufficient and necessary conditions for ground state sign-changing solutions to the Schrödinger-Poisson system with cubic nonlinearity on bounded domains, Appl. Math. Lett., 123 (2022), 107570. https://doi.org/10.1016/j.aml.2021.107570
[35]	J. Zhang, R. Niu, X. Han, Positive solutions for a nonhomogeneous Schrödinger-Poisson system, Adv. Nonlinear Anal., 11 (2022), 1201–1222. https://doi.org/10.1515/anona-2022-0238
[36]	Z. Zhang, Y. Wang, Ground state and sign-changing solutions for critical Schrödinger-Poisson system with lower order perturbation, Qual. Theory Dyn. Syst., 22, 76 (2023). https://doi.org/10.1007/s12346-023-00764-5
[37]	X. Zhong, C. 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
[38]	X. Zhong, C. Tang, Ground state sign-changing solutions for a Schrödinger-Poisson system with a $3$-linear growth nonlinearity, J. Math. Anal. Appl., 455 (2017), 1956–1974. https://doi.org/10.1016/j.jmaa.2017.04.010
[39]	W. Zou, M. Schechter, Critical point theory and its applications, Springer, New York, 2006. https://doi.org/10.1007/0-387-32968-4

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