Mathematical Notes

, Volume 103, Issue 1–2, pp 24–32 | Cite as

Nonexistence of Solutions of a Semilinear Biharmonic Equation with Singular Potential



The nonexistence of a global solution of the semilinear elliptic equation Δ2u − (C/|x|4)u − |x|σ|u|q = 0 in the exterior of a ball is studied. A sufficient condition for the nonexistence of a global solution is established. The proof is based on the test function method.


semilinear elliptic equation biharmonic operator global solution critical exponent test function method 


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  1. 1.
    É. Mitidieri and S. I. Pokhozhaev, “A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities,” in TrudyMat. Inst. Steklov (Nauka, Moscow, 2001), Vol. 234, pp. 3–383 [Proc. Steklov Inst. Math. 234, 1–362].Google Scholar
  2. 2.
    É. Mitidieri and S. I. Pokhozhaev, “Absence of global positive solutions of quasilinear elliptic inequalities,” Dokl. Ross. Akad. Nauk 359 (4), 456–460 (1998) [Dokl. Math. 57 (2), 250–253 (1998)].MathSciNetMATHGoogle Scholar
  3. 3.
    É. Mitidieri and S. I. Pokhozhaev, “Nonexistence of Positive Solutions for Quasilinear Elliptic Problems on RN,” in Trudy Mat. Inst. Steklov, Vol. 227: Studies in the Theory of Differentiable Functions of Several Variables and Its Applications. Pt. 18 (Nauka, Moscow, 1999), pp. 192–222 [Proc. Steklov Inst. Math. 227, 186–216 (1998)].Google Scholar
  4. 4.
    B. Gidas and J. Spruck, “Global and local behavior of positive solutions of linear elliptic equations,” Comm. Pure Appl. Math. 34 (4), 525–598 (1981).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    H. Brezis, Z. Dupaigne, and A. Tesei, “On a semilinear elliptic equation with inverse-square potential,” SelectaMath. (N. S.) 11 (1), 1–7 (2005).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    M. F. Bidaut-Véron and S. Pohozaev, “Nonexistence results and estimates for some nonlinear elliptic problems,” J. Anal. Math. 84, 1–49 (2001).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    J. Serrin and H. Zou, “Nonexistence of positive solutions of Lane-Emden system,” Differential Integral Equations 9 (No. 4), 635–653 (1996).MathSciNetMATHGoogle Scholar
  8. 8.
    J. Serrin, “Positive solutions of prescribedmean curvature problem,” in Calculus of Variations and Partial Differential Equations, Lect. Notes in Math. (Springer-Verlag, Berlin, 1998), Vol. 1340, pp. 248–255.Google Scholar
  9. 9.
    A. A. Kon’kov, “Behavior at infinity of solutions of second-order nonlinear equations of a particular class,” Mat. Zametki 60 (1), 30–39 (1996) [Math. Notes 60 (1–2), 22–28 (1996)].MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    A. A. Kon’kov, “On solutions of quasi-linear elliptic inequaliticontaining terms with lower-order derivatives,” Nonlinear Anal. 90, 121–134 (2013).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    A. A. Kon’kov, “On properties of solutions of quasilinear second-order elliptic inequalities,” Nonlinear Anal. 123-124, 89–114 (2015).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Sh. G. Bagyrov, “Absence of positive solutions of a second-order semilinear parabolic equation with time-periodic coefficients,” Differ. Uravn. 50 (4), 551–555 (2014) [Differ. Equations 50 (4), 548–553 (2014)].MathSciNetMATHGoogle Scholar
  13. 13.
    Sh. H. Bagyrov and M. J. Aliyev, “On absence of solutions of a semi-linear elliptic equation with biharmonic operator in the exterior of a ball,” Trans. NAS of Azerbaijan, IssueMath. 36 (4), 63–69 (2016).MathSciNetGoogle Scholar
  14. 14.
    G. G. Laptev, “On the absence of solutions for a class of singular semilinear differential inequalities,” in Trudy Mat. Inst. Steklov, Vol. 232: Function Spaces, Harmonic Analysis, and Differential Equations (Nauka, Moscow, 2001), pp. 223–235 [Proc. Steklov Inst. Math. 232, 216–228 (2001)].Google Scholar
  15. 15.
    Yu. V. Volodin, “On the critical exponents of certain nonlinear boundary-value problems with biharmonic operator in the exterior of a ball,” Mat. Zametki 79 (2), 201–212 (2006) [Math. Notes 79 (1–2), 185–195 (2006)].CrossRefMATHGoogle Scholar
  16. 16.
    Yu. V. Volodin, “The critical exponents of semilinear boundary-value problems with biharmonic operator in the exterior of a ball with boundary conditions of first type” Uchen. Zap. Ross. Gos. Sots. Univ., No. 8, 208–215 (2009).Google Scholar
  17. 17.
    X. Xu, “Uniqueness theorem for the entire positive solutions of biharmonic equations in Rn,” Proc. Roy. Soc. Edinburgh Sec. A 130 (3), 651–670 (2000).CrossRefMATHGoogle Scholar
  18. 18.
    M. Ghergu and S. D. Taliaferro, “Nonexistence of positive supersolutions of nonlinear biharmonic equations without the maximum principle,” Comm. Partial Differential Equations 40 (6), 1029–1069 (2015).MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    P. C. Carri ao, R. Demarque, and O. H. Miyagaki, “Nonlinear biharmonic problems with singular potentials,” Commun. Pure Appl. Anal. 13 (6), 2141–2154 (2014).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Y. Yao, R. Wang, and Y. Shen, “Notrivial solution for the class of semilinear biharmonic equation involving critical exponents,” ActaMath. Sci. Ser. B Engl. Ed. 27 (3), 509–514 (2007).MATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Baku State UniversityBakuAzerbaijan
  2. 2.Institute of Mathematics and MechanicsNational Academy of Sciences of AzerbaijanBakuAzerbaijan

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