Advertisement

On the extremal parameters curve of a quasilinear elliptic system of differential equations

  • Kaye Silva
  • Abiel Macedo
Article
  • 33 Downloads

Abstract

We consider a system of quasilinear elliptic equations, with indefinite super-linear nonlinearity, depending on two real parameters \(\lambda ,\mu \). By using the Nehari manifold and the notion of extremal parameter, we extend some results concerning existence of positive solutions.

Keywords

Elliptic system p-Laplacian Variational methods Extremal paramters Nehari manifold Fibering method Indefinite nonlinearity 

Mathematics Subject Classification

Primary 35G55 Secondary 35A15 35B32 35B09 

References

  1. 1.
    Alama, S., Tarantello, G.: On semilinear elliptic equations with indefinite nonlinearities. Calc. Var. Partial Differ. Equ. 1(4), 439–475 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berestycki, H., Capuzzo-Dolcetta, I., Nirenberg, L.: Variational methods for indefinite superlinear homogeneous elliptic problems. NoDEA Nonlinear Differ. Equ. Appl. 2(4), 553–572 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bobkov, V., Il’yasov, Y.: Asymptotic behaviour of branches for ground states of elliptic systems. Electron. J. Differ. Equ. 212, 1–21 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bobkov, V., Il’yasov, Y.: Maximal existence domains of positive solutions for two-parametric systems of elliptic equations. Complex Var. Elliptic Equ. 61(5), 587–607 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bozhkov, Y., Mitidieri, E.: Existence of multiple solutions for quasilinear systems via fibering method. J. Differ. Equ. 190(1), 239–267 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Drábek, P., Milota, J.: Methods of Nonlinear Analysis, 2nd edn, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser/Springer Basel AG, Basel, Applications to Differential Equations (2013)Google Scholar
  7. 7.
    Il’yasov, Y.S.: Nonlocal investigations of bifurcations of solutions of nonlinear elliptic equations. Izv. Ross. Akad. Nauk Ser. Mat. 66(6), 19–48 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ilyasov, Y.: On extreme values of nehari manifold method via nonlinear Rayleigh’s quotient. Topol. Methods Nonlinear Anal. 49(2), 683–714 (2017)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ilyasov, Y., Silva, K.: On branches of positive solutions for \(p\)-Laplacian problems at the extreme value of the Nehari manifold method. Proc. Am. Math. Soc. 146(7), 2925–2935 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lieberman, G.M.: On the natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva. Partial Differential Equations, Part 1, 2 (Warsaw, 1990), Banach Center Publications, 27, Part 1, vol. 2, Institute of Mathematics of the Polish Academy of Sciences, Warsaw, pp. 295–308 (1992)Google Scholar
  11. 11.
    Nehari, Z.: On a class of nonlinear second-order differential equations. Trans. Am. Math. Soc. 95, 101–123 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ouyang, T.: On the positive solutions of semilinear equations \(\Delta u+\lambda u+hu^p=0\) on compact manifolds. II. Indiana Univ. Math. J. 40(3), 1083–1141 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pohozaev, S.I.: The fibration method for solving nonlinear boundary value problems. Trudy Mat. Inst. Steklov. 192, 146–163 (1990). Translated in Proc. Steklov Inst. Math. 192(3), 157–173, Differential Equations and Function Spaces (Russian)Google Scholar
  14. 14.
    Silva, K., Macedo, A.: Local minimizers over the Nehari manifold for a class of concave-convex problems with sign changing nonlinearity. J. Differ. Equ. 265(5), 1894–1921 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Vázquez, J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12(3), 191–202 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Universidade Federal de GoiásGoiâniaBrazil

Personalised recommendations