Abstract
This article is devoted to study the nonlinear Schrödinger-Poisson system with pure power nonlinearities
where \(4< p<5\). By employing constraint variational method and a variant of the classical deformation lemma, we show the existence of one ground state sign-changing solution with precisely two nodal domains, which improves and generalizes the existing results by Wang, Zhang and Guan (J. Math. Anal. Appl. 479 (2019), 2284–2301).
Similar content being viewed by others
References
Alves, C.O., Souto, M.A.S., Soares, S.H.: A sign-changing solution for the Schrödinger-Poisson equation in \({\mathbb{R}}^3\). Rocky Mountain J. Math. 47, 1–25 (2017)
Ambrosetti, A.: On Schrödinger-Poisson Systems. Milan J. Math. 76, 257–274 (2008)
Ambrosetti, A., Ruiz, R.: Multiple bound states for the Schrödinger-Poisson problem. Commun. Contemp. Math. 10, 391–404 (2008)
Azzollini, A., Pomponio, A.: Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J. Math. Anal. Appl. 345, 90–108 (2008)
Batista, A.M., Furtado, M.F.: Positive and nodal solutions for a nonlinear Schrödinger-Poisson system with sign-changing potentials. Nonlinear Anal. Real World Appl. 39, 142–156 (2018)
Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. 11, 283–293 (1998)
Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486–490 (1983)
Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)
Cerami, G., Solimini, S., Struwe, M.: Some existence results for superlinear elliptic boundary value problems involving critical exponents. J. Funct. Anal. 69, 289–306 (1986)
Cerami, G., Vaira, G.: Positive solutions for some non-autonomous Schrödinger-Poisson systems. J. Differ. Equ. 248, 521–543 (2010)
Chen, S., Tang, X.: Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in \({\mathbb{R}}^3\), Z. Angew. Math. Phys., 67, 102, 18 (2016)
D’Avenia, P.: Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations. Adv. Nonlinear Stud. 2, 177–192 (2002)
Gu, L., Jin, H., Zhang, J.: Sign-changing solutions for nonlinear Schrödinger-Poisson systems with subquadratic or quadratic growth at infinity. Nonlinear Anal. 198, 111897 (2020)
Hofer, H.: Variational and topological methods in partially ordered Hilbert spaces. Math. Ann. 261, 493–514 (1982)
Huang, L., Rocha, E.M., Chen, J.: Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity. J. Math. Anal. Appl. 408, 55–69 (2013)
Ianni, I.: Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem. Topol. Methods Nonlinear Anal. 41, 365–385 (2013)
Khoutir, S.: Infinitely many high energy radial solutions for a class of nonlinear Schrödinger-Poisson system in \({\mathbb{R}}^3\). Appl. Math. Lett. 90, 139–145 (2018)
Li, G.: Some properties of weak solutions of nonlinear scalar fields equation. Ann. Acad. Sci. Fenn. Math. 14, 27–36 (1989)
Liang, Z., Xu, J., Zhu, X.: Revisit to sign-changing solutions for the nonlinear Schrödinger-Poisson system in \(\mathbb{R}^3\). J. Math. Anal. Appl. 435, 783–799 (2016)
Liu, Z., Guo, S.: On ground state solutions for the Schrödinger-Poisson equations with critical growth. J. Math. Anal. Appl. 412, 435–448 (2014)
Liu, Z., Wang, Z., Zhang, J.: Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system. Ann. Mat. Pura Appl. 195, 775–794 (2016)
Miranda, C.: Unosservazione su un teorema di Brouwer. Boll. Unione Mat. Ital. 3, 5–7 (1940)
Rabinowitz, P.H.: Variational methods for nonlinear eigenvalue problems. In: Prodi, G. (ed.) Eigenvalues of Nonlinear Problems, pp. 141–195. CIME, Odisha (1974)
Ruiz, D.: The Schrödinger-Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237, 655–674 (2006)
Ruiz, D.: On the Schrödinger-Poisson-Slater system: behavior of minimizers, radial and nonradial cases. Arch. Ration. Mech. Anal. 198, 349–368 (2010)
Sánchez, O., Soler, J.: Long-time dynamics of the Schrödinger-Poisson-Slater system. J. Stat. Phys. 114, 179–204 (2004)
Shuai, W., Wang, Q.: Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger-Poisson system in \({\mathbb{R}}^3\). Z. Angew. Math. Phys. 66, 3267–3282 (2015)
Tarantello, G.: Nodal solutions of semilinear elliptic equations with critical exponent. Differ. Integral Equ. 5, 25–42 (1992)
Wang, D., Zhang, H., Guan, W.: Existence of least-energy sign-changing solutions for Schrödinger-Poisson system with critical growth. J. Math. Anal. Appl. 479, 2284–2301 (2019)
Wang, Z., Zhou, H.: Sign-changing solutions for the nonlinear Schrödinger-Poisson system in \({\mathbb{R}}^3\). Calc. Var. Partial Differential Equations 52, 927–943 (2015)
Willem, M.: Minimax Theorems. Birkhäuser Boston Inc, Boston (1996)
Zhang, J.: On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth. Nonlinear Anal. 75, 6391–6401 (2012)
Zhang, J.: On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth. J. Math. Anal. Appl. 428, 387–404 (2015)
Zhang, J.: Ground state and multiple solutions for Schrödinger-Poisson equations with critical nonlinearity. J. Math. Anal. Appl. 440, 466–482 (2016)
Zhao, L., Zhao, F.: On the existence of solutions for the Schrödinger-Poisson equations. J. Math. Anal. Appl. 346, 155–169 (2008)
Zhong, X., Tang, C.: Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in \({\mathbb{R}}^3\). Nonlinear Anal. Real World Appl. 39, 166–184 (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhang, Z., Wang, Y. & Yuan, R. Ground State Sign-Changing Solution for Schrödinger-Poisson System with Critical Growth. Qual. Theory Dyn. Syst. 20, 48 (2021). https://doi.org/10.1007/s12346-021-00487-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-021-00487-5