A note on the nonexistence of positive supersolutions to elliptic equations with gradient terms


We prove that if the elliptic problem \(-\Delta u+b(x)|\nabla u|=c(x)u\) with \(c\ge 0\) has a positive supersolution in a domain \(\varOmega \) of \( {\mathrm {R}}^{N\ge 3}\), then cb must satisfy the inequality

$$\begin{aligned} \sqrt{ \int _\varOmega c\phi ^2}\le \sqrt{ \int _\varOmega | \nabla \phi |^2}+\sqrt{ \int _\varOmega \frac{b^2}{4}\phi ^2},\quad \phi \in C_c^\infty (\varOmega ). \end{aligned}$$

As an application, we obtain Liouville-type theorems for positive supersolutions in exterior domains when \(c(x)-\frac{b^2(x)}{4}>0\) for large |x|, but unlike the known results, we allow the case \(\lim _{|x|\rightarrow \infty }c(x)-\frac{b^2(x)}{4}=0\). The weights b and c are allowed to be unbounded. In particular, among other things, we show that if \(\tau :=\limsup _{|x| \rightarrow \infty }|xb(x)|<\infty, \) then this problem does not admit any positive supersolution if

$$\begin{aligned} \liminf _{|x| \rightarrow \infty }|x|^2c(x)> \frac{(N-2+\tau )^2}{4}, \end{aligned}$$

and, when \(\tau =\infty , \) we have the same if

$$\begin{aligned} \limsup _{R\rightarrow \infty } R\left( \frac{ \inf _{R<|x|<2 R} (c(x)-\frac{b(x)^2}{4})}{\sup _{\frac{R}{2}<|x|<4 R}|b(x)|}\right) =\infty . \end{aligned}$$

This is a preview of subscription content, access via your institution.


  1. 1.

    Agmon, S.: On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds. In: Methods of Functional Analysis and Theory of Elliptic Equations, Liguori, Naples, pp. 19–52 (1983)

  2. 2.

    Armstrong, S.N., Sirakov, B.: Nonexistence of positive supersolutions of elliptic equations via the maximum principle. Commun. Partial Differ. Equ. 36, 2011–2047 (2011)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Alarcon, S., Garcia-Melian, J., Quaas, A.: Liouville type theorems for elliptic equations with gradient terms. Milan J. Math. 13(81), 171–185 (2013)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Alarcon, S., Garcia-Melian, J., Quaas, A.: Nonexistence of positive supersolutions to some nonlinear elliptic problems. J. Math. Pures Appl. 99, 618–634 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Alarcon, S., Garcia-Melian, J., Quaas, A.: Existence and non-existence of solutions to elliptic equations with a general convection term. Proc. R. Soc. Edinb. 144A, 225–239 (2014)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Alarcon, S., Garcia-Melian, J., Quaas, A.: Keller–Osserman type conditions for some elliptic problems with gradient terms. J. Differ. Equ. 252, 886–914 (2012)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Arcoya, D., De Coster, C., Jeanjean, L., Tanaka, K.: Continuum of solutions for an elliptic problem with critical growth in the gradient. J. Funct. Anal. 268(8), 2298–2335 (2015)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Berestycki, H., Hamel, F., Nadirashvili, N.: The speed of propagation for KPP type problems I. Periodic framework. J. Eur. Math. Soc. 7, 173–213 (2005)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Berestycki, H., Hamel, F., Rossi, L.: Liouville type results for semilinear elliptic equations in unbounded domains. Ann. Mat. Pura Appl. (4) 186, 469–507 (2007)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Burgos-Perez, M.A., Garcia Melian, J., Quaas, A.: Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems. Discrete Contin. Dyn. Syst. (2016). https://doi.org/10.3934/dcds.2016004

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Capuzzo Dolcetta, I., Cutri, A.: Hadamard and Liouville type results for fully nonlinear partial differential inequalities. Commun. Contemp. Math. 5(3), 435–448 (2003)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Caristi, G., Mitidieri, E.: Nonexistence of positive solutions of quasilinear equations. Adv. Differ. Equ. 2, 317–359 (1997)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Chen, H., Felmer, P.: On Liouville type theorems for fully nonlinear elliptic equations with gradient term. J. Differ. Equ. 255, 2167–2195 (2013)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Chen, H., Peng, R., Zhou, F.: Nonexistence of Positive Supersolution to a Class of Semilinear Elliptic Equations and Systems in an Exterior Domain. arXiv:1803.02531

  15. 15.

    Chen, H., Quaas, A., Zhou, F.: On nonhomogeneous elliptic equations with the Hardy–Leray potentials. arXiv:1705.08047

  16. 16.

    Cowan, C.: Optimal Hardy inequalities for general elliptic operators with improvements. Commun. Pure Appl. Anal. 9(1), 109–140 (2010)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Devyver, B., Fraas, M., Pinchover, Y.: Optimal hardy weight for second-order elliptic operator: an answer to a problem of Agmon. J. Funct. Anal. 266, 4422–4489 (2014)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Felmer, P., Quaas, A., Sirakov, B.: Solvability of nonlinear elliptic equations with gradient terms. J. Differ. Equ. 254(11), 4327–4346 (2013)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Jeanjean, L., Sirakov, B.: Existence and multiplicity for elliptic problems with quadratic growth in the gradient. Commun. Partial Differ. Equ. 38, 244–264 (2013)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Pinchover, Y.: A Liouville-type theorem for Schrödinger operators. Commun. Math. Phys. 272(1), 75–84 (2007)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Pinchover, Y.: Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations. In: Proceedings of Symposia in Pure Mathematics, vol. 76, no. 1, pp. 329–356. American Mathematical Society, Providence (2007)

  22. 22.

    Rossi, L.: Non-existence of positive solutions of fully nonlinear elliptic bounded domains. Commun. Pure Appl. Anal. 7, 125–141 (2008)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Serrin, J., Zou, H.: Cauchy–Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta. Math. 189, 79–142 (2002)

    MathSciNet  Article  Google Scholar 

Download references


The authors thank the referees for their valuable suggestions to improve the presentation of the original manuscript. A. Aghajani was partially supported by Grant from IPM (No. 99350212). C. Cowan supported in part by NSERC

Author information



Corresponding author

Correspondence to A. Aghajani.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aghajani, A., Cowan, C. A note on the nonexistence of positive supersolutions to elliptic equations with gradient terms. Annali di Matematica 200, 125–135 (2021). https://doi.org/10.1007/s10231-020-00987-2

Download citation


  • Liouville-type theorems
  • Supersolutions
  • Gradient term

Mathematics Subject Classification

  • 35J60
  • 35B53