Liouville type theorems for stable solutions of the weighted elliptic system with the advection term: \( \varvec{p \ge \vartheta >1}\)



In this paper, we are concerned with the weighted elliptic system with the advection term
$$\begin{aligned} {\left\{ \begin{array}{ll} -\omega (x)\Delta u(x)-\nabla \omega (x)\cdot \nabla u(x)=\omega _1 v^{\vartheta },\\ -\omega (x)\Delta v(x)-\nabla \omega (x)\cdot \nabla v(x)=\omega _2 u^p, \end{array}\right. } \quad \text{ in }\;\ \mathbb {R}^N, \end{aligned}$$
where \(N \ge 3\), \(p \ge \vartheta >1\) and \(\omega , \omega _1, \omega _2 \ne 1\) satisfy some suitable conditions. We establish Liouville type theorems for stable solutions with two cases \(\omega _1 \ne \omega _2\) and \(\omega _1 \equiv \omega _2\), respectively. Based on a delicate application of some new techniques, these difficulties caused by the advection term and the weighted term are overcome, and the sharp results are obtained.


Weighted elliptic system Advection term Stable solutions Liouville type theorem 

Mathematics Subject Classification

35B33 35B45 35B53 35J60 



The author would wish to express their appreciations to the anonymous referee for his/her valuable suggestions, which have greatly improved this paper. The author wishes to express his warmest thanks to Professor Tianling Jin for his hospitality and Department of Mathematics, Hong Kong University of Science and Technology (where part of this work was done). The work was partially supported by NSFC of China (No. 11471174), Natural Science Foundation of Zhejiang Province (No. LY17A010007) and K.C.Wong Magna Fund in Ningbo University.


  1. 1.
    Ai, S.B., Cowan, C.: Perturbations of Lane–Emden and Hamilton–Jacobi equations II: exterior domains. J. Differ. Equ. 260, 8025–8050 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ai, S.B., Cowan, C.: Perturbations of Lane–Emden and Hamilton–Jacobi equations: the full space case. Nonlinear Anal. 151, 227–251 (2017)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cowan, C.: Liouville theorems for stable Lane–Emden systems with biharmonic problems. Nonlinearity 26, 2357–2371 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cowan, C., Fazly, M.: On stable entire solutions of semi-linear elliptic equations with weights. Proc. Am. Math. Soc. 140, 2003–2012 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cowan, C., Ghoussoub, N.: Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains. Calc. Var. PDEs 49, 291–305 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dancer, E.N., Du, Y.H., Guo, Z.M.: Finite Morse index solutions of an elliptic equation with supercritical exponent. J. Differ. Equ. 250, 3281–3310 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dávila, J., Dupaigne, L., Wang, K.L., Wei, J.C.: A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem. Adv. Math. 258, 240–285 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dupaigne, L., Ghergu, M., Goubet, O., Warnault, G.: The Gel’fand problem for the biharmonic operator. Arch. Ration. Mech. Anal. 208, 725–752 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Farina, A.: On the classification of solutions of Lane–Emden equation on unbounded domains of \(\mathbb{R}^N\). J. Math. Pures Appl. 87, 537–561 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fazly, M.: Liouville type theorems for stable solutions of certain elliptic systems. Adv. Nonlinear Stud. 12, 1–17 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fazly, M., Wei, J.C.: On stable solutions of the fractional Hénon–Lane–Emden equation. Commun. Contemp. Math. 18, 1650005 (2016)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Guo, Z.M., Wei, J.C.: Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents. Discrete Contin. Dyn. Syst. 34, 2561–2580 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hajlaoui, H., Harrabi, A., Mtiri, F.: Liouville theorems for stable solutions of the weighted Lane–Emden system. Discrete Contin. Dyn. Syst. 37, 265–279 (2017)MathSciNetMATHGoogle Scholar
  14. 14.
    Hajlaoui, H., Harrabi, A., Ye, D.: On stable solutions of biharmonic problems with polynomial growth. Pacific J. Math. 270, 79–93 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hu, L.G.: Liouville type results for semi-stable solutions of the weighted Lane–Emden system. J. Math. Anal. Appl. 432, 429–440 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hu, L.G.: A monotonicity formula and Liouville-type theorems for stable solutions of the weighted elliptic system. Adv. Differ. Equ. 22, 49–76 (2017)MathSciNetMATHGoogle Scholar
  17. 17.
    Mitidieri, E., Pohozaev, S.I.: A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities. Tr. Mat. Inst. Steklova 234, 1–384 (2001)MathSciNetMATHGoogle Scholar
  18. 18.
    Montenegro, M.: Minimal solutions for a class of elliptic systems. Bull. Lond. Math. Soc. 37, 405–416 (2005)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Pacard, F.: Partial regularity for weak solutions of a nonlinear elliptic equations. Manuscripta Math. 79, 161–172 (1993)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Phan, Q.H.: Liouville-type theorems and bounds of solutions for Hardy-Hénon elliptic systems. Adv. Differ. Equ. 17, 605–634 (2012)MATHGoogle Scholar
  21. 21.
    Serrin, J., Zou, H.H.: Nonexistence of positive solutions of Lane–Emden systems. Differ. Integral Equ. 9, 635–653 (1996)MATHGoogle Scholar
  22. 22.
    Souplet, P.: The proof of the Lane–Emden conjecture in four space dimensions. Adv. Math. 221, 1409–1427 (2009)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Wang, K.L.: Partial regularity of stable solutions to the supercritical equations and its applications. Nonlinear Anal. 75, 5238–5260 (2012)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Wei, J.C., Ye, D.: Liouville theorems for stable solutions of biharmonic problem. Math. Ann. 356, 1599–1612 (2013)MathSciNetCrossRefMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNingbo UniversityNingboPeople’s Republic of China

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