Multiple positive bound states for critical Schrödinger-Poisson systems

Cerami, Giovanna; Molle, Riccardo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 73.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

Using variational methods we prove some results about existence and multiplicity of positive bound states of to the following Schrödinger-Poisson system: $$

We remark that (SP) exhibits a “double” lack of compactness because of the unboundedness of ℝ³ and the critical growth of the nonlinear term and that in our assumptions ground state solutions of (SP) do not exist.

[1] A. Ambrosetti, On Schrödinger-Poisson systems. Milan J. Math. 10 (2008) 391–404. | MR

[2] V. Benci and G. Cerami, Existence of positive solutions of the equation − Δu + a(x)u = u^{(N+2)∕(N−2)} in ℝ^N. J. Funct. Anal. 88 (1990) 90–117. | DOI | MR | Zbl

[3] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations. Top. Methods Nonlin. Anal. 11 (1998) 283–293. | MR | Zbl

[4] V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein–Gordon equation coupled with Maxwell equations. Rev. Math. Phys. 14 (2002) 409–420. | DOI | MR | Zbl

[5] V. Benci and D. Fortunato, Solitons in Schrödinger-Maxwell equations. J. Fixed Point Theory Appl. 15 (2014) 101–132. | DOI | MR | Zbl

[6] V. Benci and D. Fortunato, Variational methods in nonlinear field equations. Springer Monographs in Math. Springer Int. Publ. Switzerland (2014). | DOI | MR | Zbl

[7] R. Benguria, H. Brézis and E. Lieb, The Thomas-Fermi-Von Weizsäcker theory of atoms and molecules. Commun. Math. Phys. 79 (1981) 167–180. | DOI | MR | Zbl

[8] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36 (1983) 437–477. | DOI | MR | Zbl

[9] I. Catto and P.L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. I. Commun Partial Differ. Equ. 17 (1992) 1051–1110. | MR | Zbl

[10] J. Chen, E. Rocha and L. Huang, Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity. J. Math. Anal. Appl. 408 (2013) 55–69. | DOI | MR | Zbl

[11] G. Cerami and R. Molle, Multiple positive solutions for nonautonomous quasicritical elliptic problems in unbounded domains. Adv. Nonlin. Stud. 6 (2006) 233–254. | DOI | MR | Zbl

[12] G. Cerami and R. Molle, Positive bound state solutions for some Schrödinger-Poisson systems. Nonlinearity 29 (2016) 3103–3119. | DOI | MR | Zbl

[13] G. Cerami and D. Passaseo, High energy positive solutions for mixed and Neumann elliptic problems with critical nonlinearity. J. Anal. Math. 71 (1997) 1–39. | DOI | MR | Zbl

[14] G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents. J. Funct. Anal. 69 (1986) 289–306. | DOI | MR | Zbl

[15] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems. J. Differ. Equ. 248 (2010) 521–543. | DOI | MR | Zbl

[16] W-Y. Ding, On a conformally invariant elliptic equation on ℝⁿ. Commun. Math. Phys. 107 (1986) 331–335. | DOI | MR | Zbl

[17] B. Gidas, Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations, in Proc of Nonlinear Partial Differential Equations In Engineering and Applied Science, edited by R.L Sternberg, A.J Kalinowski, J.S Papadakis. Dekker, New York (1979). | MR | Zbl

[18] B. Gidas, W.M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in ℝⁿ. Math. Anal. Appl. A: Adv. Math. Suppl. Stud. 7A (1981) 369–402. | MR | Zbl

[19] Y. He, L. Lu and W. Shuai, Concentrating ground state solutions of Schrödinger-Poisson equations in ℝ³ involving critical Sobolev exponents. Commun. Pure Appl. Anal. 15 (2016) 103–125. | MR | Zbl

[20] X. He and W. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth. J. Math. Phys. 53 (2012) 023702. | DOI | MR | Zbl

[21] P.L. Lions, Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys. 109 (1987) 33–97. | DOI | MR | Zbl

[22] P.L. Lions and B. Simon, The Thomas Fermi theory of atoms, molecules and solids. Adv. Math. 23 (1977) 22–116. | DOI | MR | Zbl

[23] Z. Liu and S. Guo, On ground state solutions for the Schrödinger-Poisson equations with critical growth. J. Math. Anal. Appl. 412 (2014) 435–448. | DOI | MR | Zbl

[24] Z. Liu, S. Guo and Y. Fang, Multiple semi classical states for Schrödinger-Poisson equations with critical exponential growth. J. Math. Phys. 56 (2015) 041505. | DOI | MR | Zbl

[25] P.A. Markowich, C.A. Ringhofer and C. Schmeiser, Semiconductor Equations. Springer Velag, Vienna (1990). | DOI | MR | Zbl

[26] D. Passaseo, Some sufficient conditions for the existence of positive solutions to the equation − Δu + a(x)u = u^2*−1 in bounded domains. Ann. Inst. Henri Poincaré Anal. Non Linéaire 13 (1996) 185–227. | DOI | Numdam | MR | Zbl

[27] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinearlocal term. J. Funct. Anal. 237 (2006) 655–674. | DOI | MR | Zbl

[28] G. Talenti, Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976) 353–372. | DOI | MR | Zbl

[29] J. Zhang, On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth. J. Math. Anal. Appl. 428 (2015) 387–404. | DOI | MR | Zbl

[30] J. Zhang, Ground state and multiple solutions for Schrödinger-Poisson equations with critical nonlinearity. J. Math. Anal. Appl. 440 (2016) 466–482. | DOI | MR | Zbl