Non-radial solutions to a bi-harmonic equation with negative exponent

  • Ali HyderEmail author
  • Juncheng Wei


We prove the existence of smooth non-radial entire solution to
$$\begin{aligned} \Delta ^2 u+u^{-q}=0\quad \text {in }\mathbb {R}^3,\quad u>0, \end{aligned}$$
for \(q>1\). This answers an open question raised by McKenna and Reichel (Electron J Differ Equ 37:1–3, 2003).

Mathematics Subject Classification

35J61 35B40 53A30 



  1. 1.
    Branson, T.P.: Group representations arising from Lorentz conformal geometry. J. Funct. Anal. 74, 199–291 (1987)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Caffarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42, 271–297 (1989)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chang, S.-Y.A., Chen, W.: A note on a class of higher order conformally covariant equations. Discrete Contin. Dyn. Syst. 63, 275–281 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, W., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63, 615–622 (1991)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Choi, Y.S., Xu, X.: Nonlinear biharmonic equations with negative exponents. J. Differ. Equ. 246, 216–234 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Duoc, T.V., Ngô, Q.A.: A note on positive radial solutions of \(\Delta ^2 u+u^{-q}=0\) in \({\mathbb{R}}^3\) with exactly quadratic growth at infinity. Differ. Integral Equ. 30(11–12), 917–928 (2017)zbMATHGoogle Scholar
  7. 7.
    Feng, X., Xu, X.: Entire solutions of an integral equation in \({\mathbb{R}}^5\). ISRN Math. Anal. p. 17 (2013).
  8. 8.
    Guerra, I.: A note on nonlinear biharmonic equations with negative exponents. J. Differ. Equ. 253, 3147–3157 (2012) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hyder, A.: Conformally Euclidean metrics on \({\mathbb{R}}^n\) with arbitrary total Q-curvature. Anal. PDE 10(3), 635–652 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hyder, A., Martinazzi, L.: Conformal metrics on \(\mathbb{R}^{2m}\) with constant \(Q\)-curvature, prescribed volume and asymptotic behavior. Discrete Contin. Dyn. Syst. A. 35(1), 283–299 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lai, B.: A new proof of I. Guerra’s results concerning nonlinear biharmonic equations with negative exponents. J. Math. Anal. Appl. 418, 469–475 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Li, Y.: Remarks on some conformally invariant integral equations: the method of moving spheres. J. Eur. Math. Soc. 6, 1–28 (2004)MathSciNetGoogle Scholar
  13. 13.
    Lin, C.S.: A classification of solutions of a conformally invariant fourth order equation in \(\mathbb{R}^n\). Comment. Math. Helv. 73, 206–231 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Martinazzi, L.: Classification of solutions to the higher order Liouville’s equation on \(\mathbb{R}^{2m}\). Math. Z. 263, 307–329 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Martinazzi, L.: Conformal metrics on \({\mathbb{R}}^{2m}\) with constant Q-curvature and large volume. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(6), 969–982 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    McKenna, P.J., Reichel, W.: Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry. Electron. J. Differ. Equ. 37, 1–13 (2003)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ngô, Q.A.: Classification of entire solutions of \((-\Delta )^n u+u^{4n-1}=0\) with exact linear growth at infinity in \(\mathbb{R}^{2n-1}\). Proc. Am. Math. Soc. 146(6), 2585–2600 (2018)CrossRefGoogle Scholar
  18. 18.
    Pohozaev, S.: Eigenfunctions of the equation \(\Delta u+\lambda f(u)=0\). Sov. Math. Dokl. 6, 1408–1411 (1965)Google Scholar
  19. 19.
    Wei, J., Xu, X.: Classification of solutions of higher order conformally invariant equations. Math. Ann. 313(2), 207–228 (1999)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wei, J., Ye, D.: Nonradial solutions for a conformally invariant fourth order equation in \({\mathbb{R}}^4\). Calc. Var. Partial Differ. Equ. 32(3), 373–386 (2008)CrossRefGoogle Scholar
  21. 21.
    Xu, X.: Exact solutions of nonlinear conformally invariant integral equations in \(\mathbb{R}^3\). Adv. Math. 194, 485–503 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Yang, P., Zhu, M.: On the Paneitz energy on standard three sphere. ESAIM Control Optim. Calc. Var. 10(2), 211–223 (2004)MathSciNetCrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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