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Ground state solutions for Schrödinger–Poisson systems with multiple weighted critical exponents

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Abstract

In this paper we study the existence of nonnegative radial ground state solutions and nontrivial radial solutions of a Schrödinger–Poisson system having weights of power type in both Schrödinger term and the Poisson term. The nonlinearity may involve with single or multiple weighted critical exponents. The main abstract methods we use are the Ekeland variational principle, the Nehari manifold method and the mountain pass theorem.

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References

  1. Alves, C.O., Souto, M.A.S.: Existence of least energy nodal solution for a Schrödinger–Poisson system in bounded domains. Z. Angew. Math. Phys. 65, 1153–1166 (2014)

    Article  MathSciNet  Google Scholar 

  2. Ambrosetti, A.: On Schrödinger–Poisson Systems. Milan J. Math. 76, 257–274 (2008)

    Article  MathSciNet  Google Scholar 

  3. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)

    Article  MathSciNet  Google Scholar 

  4. Ambrosetti, A., Ruiz, R.: Multiple bound states for the Schrödinger–Poisson problem. Commun. Contemp. Math. 10, 391–404 (2008)

    Article  MathSciNet  Google Scholar 

  5. Azzollini, A., Pomponio, A.: Ground state solutions for the nonlinear Schrödinger–Maxwell equations. J. Math. Anal. Appl. 345, 90–108 (2008)

    Article  MathSciNet  Google Scholar 

  6. Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger–Maxwell equations. Topol. Methods Nonlinear Anal. 11, 283–293 (1998)

    Article  MathSciNet  Google Scholar 

  7. Benci, V., Fortunato, D.: Solitary waves of the nonlinear Klein–Gordon equation coupled with the Maxwell equations. Rev. Math. Phys. 14, 409–420 (2002)

    Article  MathSciNet  Google Scholar 

  8. Bonheure, D., Mercuri, C.: Embedding theorems and existence results for nonlinear Schrödinger–Poisson systems with unbounded and vanishing potentials. J. Differ. Equ. 251, 1056–1085 (2011)

    Article  Google Scholar 

  9. Benguria, R., Brézis, H., Lieb, E.H.: The Thomas–Fermi–von Weizsäcker theory of atoms and molecules. Commun. Math. Phys. 79, 167–180 (1981)

    Article  Google Scholar 

  10. Brézis, H., Lieb, E.H.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486–490 (1983)

    Article  MathSciNet  Google Scholar 

  11. Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)

    Article  MathSciNet  Google Scholar 

  12. Catto, I., Lions, P.L.: Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. I. A necessary and sufficient condition for the stability of general molecular system. Commun. Partial Differ. Equ. 17, 1051–1110 (1992)

    Article  MathSciNet  Google Scholar 

  13. Cerami, G., Vaira, G.: Positive solutions for some non-autonomous Schrödinger–Poisson systems. J Differ. Equ. 248, 521–543 (2010)

    Article  Google Scholar 

  14. d’Avenia, P., Pomponio, A., Vaira, G.: Infinitely many positive solutions for a Schrödinger–Poisson system. Nonlinear Anal. 74, 5705–5721 (2011)

  15. D’Aprile, T., Mugnai, D.: Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations. Proc. Roy. Soc. Edinb. Sect. A 134, 893–906 (2004)

  16. D’Aprile, T., Mugnai, D.: Non-existence results for the coupled Klein–Gordon–Maxwell equations. Adv. Nonlinear Stud. 4, 307–322 (2004)

  17. Du, Y., Su, J.: Ground state solutions for Schrödinger–Poisson systems involving a weighted critical exponent. Preprint

  18. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MathSciNet  Google Scholar 

  19. Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 24, 525–598 (1981)

    Article  MathSciNet  Google Scholar 

  20. Gladiali, F., Grossi, M., Neves, S.L.N.: Nonradial solutions for the Hénon equation in \({\mathbb{R}}^N\). Adv. Math. 249, 1–36 (2013)

    Article  MathSciNet  Google Scholar 

  21. Hénon, M.: Numerical experiments on the stability of spherical stellar systems. Astron. Astrophys. 24, 229–238 (1973)

    Google Scholar 

  22. Ianni, I., Ruiz, D.: Ground and bound states for a static Schrödinger–Poisson–Slater problem. Commun. Contemp. Math. 14, 1250003 (2012). (22 pages)

    Article  MathSciNet  Google Scholar 

  23. Iturriaga, L., Moreira dos Santos, E., Ubilla, P.: Local minimizers in spaces of symmetric functions and applications. J. Math. Anal. Appl. 429, 27–56 (2015)

    Article  MathSciNet  Google Scholar 

  24. Li, A., Su, J., Zhao, L.: Existence and multiplicity of solutions of Schrödinger–Poisson systems with radial potentials. Proc. R. Soc. Edinb. Sect. A 144, 319–332 (2014)

    Article  Google Scholar 

  25. Lieb, E.H.: Thomas-Fermi and related theories and molecules. Rev. Modern Phys. 53, 603–641 (1981)

    Article  MathSciNet  Google Scholar 

  26. Lieb, E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. 118, 349–374 (1983)

    Article  MathSciNet  Google Scholar 

  27. Lions, P.L.: Solutions of Hartree–Fock equations for Coulomb systems. Commun. Math. Phys. 109, 33–97 (1987)

    Article  MathSciNet  Google Scholar 

  28. Markowich, P., Ringhofer, C., Schmeiser, C.: Semiconductor Equations. Springer, New York (1990)

    Book  Google Scholar 

  29. Mugnai, D.: The Schrödinger–Poisson system with positive potential. Commun. Partial Differ. Equ. 36, 1099–1117 (2011)

    Article  Google Scholar 

  30. Mercuri, C.: Positive solutions of nonlinear Schrödinger–Poisson systems with radial potentials vanishing at infinity. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 19, 211–227 (2008)

    Article  MathSciNet  Google Scholar 

  31. Mercuri, C., Moroz, V., Van Schaftingen, J.: Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency. Calc. Var. Partial Differ. Equ. 55 (2016). Art.146, 58 pp

  32. Ni, W.M.: A nonlinear Dirichlet problem on the unit ball and its applications. Indiana Univ. Math. J. 31, 801–807 (1982)

    Article  MathSciNet  Google Scholar 

  33. Ruiz, D.: The Schrödinger–Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237, 655–674 (2006)

    Article  MathSciNet  Google Scholar 

  34. Ruiz, D.: On the Schrödinger–Poisson–Slater system: behavior of minimizers, radial and nonradial cases. Arch. Ration. Mech. Anal. 198, 349–368 (2010)

    Article  MathSciNet  Google Scholar 

  35. Smets, D., Su, J., Willem, M.: Nonradial ground states for the Hénon equation. Commun. Contemp. Math. 4, 467–480 (2002)

    Article  MathSciNet  Google Scholar 

  36. Su, J., Tian, R.: Weighted Sobolev type embedding and coercive quasilinear elliptic equations on \({\mathbb{R}}^N\). Proc. Am. Math. Soc. 140, 891–903 (2012)

    Article  Google Scholar 

  37. Su, J., Wang, Z.-Q., Willem, M.: Nonlinear Schrödinger equations with unbounded and decaying radial potentials. Commun. Contemp. Math. 9, 571–583 (2007)

    Article  MathSciNet  Google Scholar 

  38. Su, J., Wang, Z.-Q., Willem, M.: Weighted Sobolev embedding with unbounded and decaying radial potential. J. Differ. Equ. 238, 201–219 (2007)

    Article  MathSciNet  Google Scholar 

  39. Sun, M., Su, J., Zhao, L.: Infinitely many solutions for a Schrödinger–Poisson system with concave and convex nonlinearities. Discrete Contin. Dyn. Syst. 35, 427–440 (2015)

    Article  MathSciNet  Google Scholar 

  40. Sun, M., Su, J., Zhao, L.: Solutions of a Schrödinger–Poisson system with combined nonlinearities. J. Math. Anal. Appl. 442, 385–403 (2016)

    Article  MathSciNet  Google Scholar 

  41. Wang, C., Su, J.: Positive radial solutions of critical Hénon equations on the unit ball in \({\mathbb{R}}^N\). Preprint (2019)

  42. Wang, C., Su, J.: The existence of ground state solutions for critical Hénon equation in \({\mathbb{R}}^N\). Preprint (2019)

  43. Wang, C., Su, J.: Critical exponents of weighted Sobolev embeddings for radial functions. Appl. Math. Lett. 107, 106484 (2020). (6 pp)

    Article  MathSciNet  Google Scholar 

  44. Willem, M.: Minimax Theorems. Birkhäuser Boston. Inc., Boston (1996)

    Book  Google Scholar 

  45. Zhao, L., Zhao, F.: On the existence of solutions for the Schrödinger–Poisson equations. J. Math. Anal. Appl. 346, 155–169 (2008)

    Article  MathSciNet  Google Scholar 

  46. Zhao, L., Zhao, F.: Positive solutions for Schrödinger–Poisson equations with a critical exponent. Nonlinear Anal. 70, 2150–2164 (2009)

    Article  MathSciNet  Google Scholar 

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  1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, People’s Republic of China

    Yao Du & Jiabao Su

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  1. Yao Du
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  2. Jiabao Su
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Correspondence to Jiabao Su.

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Du, Y., Su, J. Ground state solutions for Schrödinger–Poisson systems with multiple weighted critical exponents. Nonlinear Differ. Equ. Appl. 28, 66 (2021). https://doi.org/10.1007/s00030-021-00728-1

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