Global integrability for solutions to quasilinear elliptic systems


This paper deals with global integrability for solutions to quasilinear elliptic systems involving N equations of the form

$$\begin{aligned} \left\{ \begin{array}{llll} \displaystyle -\sum _{i=1}^N \frac{\partial }{\partial x_i} \left( \sum _{\beta =1}^N \sum _{j=1}^n a_{i,j}^{\alpha ,\beta } (x,u(x))\frac{\partial u^\beta (x)}{\partial x_j} \right) =f^\alpha (x), &{} x\in \Omega , \\ \displaystyle u(x)=0, &{} x\in \partial \Omega , \end{array}\right. \end{aligned}$$

where \(\alpha \in \{1,\ldots ,N\}\) is the equation index, \(\Omega \) is an open bounded subset of \({\mathbb {R}}^n\), \(n>2\), \(u=(u^1,\ldots ,u^N): \Omega \rightarrow {\mathbb {R}}^N\) and \(f\in L^m(\Omega ), \frac{2n}{n+2}\le m\le \frac{n}{2}\). Under ellipticity condition of diagonal coefficients, smallness and staircase support conditions of off-diagonal coefficients, we derive some global integrability results.

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


  1. 1.

    Boccardo, L.: Problemi differenziali ellittici e parabolici con dati misure. Boll. Unione Mat. Ital. Sez. A 11, 439–461 (1997)

    Google Scholar 

  2. 2.

    Baroni, P.: Riesz potential estimates for a general class of quasilinear equations. Calc. Var. Partial Differ. Equ. 53, 803–846 (2015)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Benilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vazquez, J.L.: An \(L^1\) theory of existence and uniqueness of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22, 240–273 (1995)

    MATH  Google Scholar 

  4. 4.

    Boccardo, L., Croce, G.: Elliptic Partial Differential Equations, De Gruyter Studies in Mathematics, vol. 55. De Gruyter, Berlin (2014)

    Google Scholar 

  5. 5.

    Boccardo, L., Croce, G.: Esistenza e regolarità di soluzioni di alcuni problemi ellittici, In: Quaderni dell\(^{\prime }\)UMI, vol. 51, Pitagora Editrice, Bologna (2010)

  6. 6.

    Boccardo, L., Cirmi, G.R.: Existence and uniqueness of solution of unilateral problems with \(L^1\) data. J. Convex Anal. 6, 195–206 (1999)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Boccardo, L., Dall’Aglio, A., Orsina, L.: Existence and regularity results for some elliptic equations with degenerate coercivity. Atti Sem. Mat. Fis. Univ. Modena. 46, 51–81 (1998)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Boccardo, L., Gallouët, T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Boccardo, L., Gallouët, T., Orsina, L.: Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. H. Poincaré Anal. Non Linéaire 13, 539–551 (1996)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Betta, M.F., Del Vecchio, T., Posteraro, M.R.: Existence and regularity results for nonlinear degenerate elliptic equations with measure data. Ric. Mat. 47, 277–295 (1998)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Boccardo, L., Gallouët, T., Marcellini, P.: Anisotropic equations in \(L^1\). Differ. Integral Equ. 9, 209–212 (1996)

    MATH  Google Scholar 

  12. 12.

    Cirmi, G.R.: On the existence of solutions to nonlinear degenerate elliptic equations with measure data. Ric. Mat. 42, 315–329 (1993)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Cirmi, G.R., Leonardi, S.: Regularity results for the gradient of solutions linear elliptic equations with \(L^{1,\lambda }\) data. Ann. Mat. Pura Appl. 185, 537–553 (2006)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Dall’Aglio, A.: Approximated solutions of equations with \(L^1\) data. Application to the \(H\)-convergence of quasi-linear parabolic equations. Ann. Mat. Pura Appl 170, 207–240 (1996)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Dolzmann, G., Hungerbühler, N., Müller, S.: The \(p\)-harmonic systems with measure-valued right hand side. Ann. Inst. H. Poincaré Anal. Non Linéaire 14, 353–364 (1997)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Fuchs, M., Reuling, J.: Non-linear elliptic systems involving measure data. Rend. Mat. Appl. 15, 101–109 (1995)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Gao, H., Liang, S., Cui, Y.: Regularity for anisotropic solutions to some nonlinear elliptic system. Front. Math. China 11, 77–87 (2016)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Gao, H., Leonetti, F., Ren, W.: Regularity for anisotropic elliptic equations with degenerate coercivity. Nonlinear Anal. 187, 493–505 (2019)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Kovalevskii, A.A.: On the summability of entropy solutions for the Dirichlet problem in a class of non-linear elliptic fourth-order equations. Izv. Math. 67, 881–894 (2003)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Kuusi, T., Mingione, G.: Universal potential estimates. J. Funct. Anal. 262, 4205–4269 (2012)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Leonardi, S., Leonetti, F., Pignotti, C., Rocha, E., Staicu, V.: Maximum principles for some quasilinear elliptic systems. Nonlinear Anal.

  22. 22.

    Leone, C., Porretta, A.: Entropy solutions to nonlinear elliptic equations in \(L^1\). Nonlinear Anal. 32, 325–334 (1998)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Leonetti, F., Petricca, P.V.: Anisotropic elliptic systems with measure data. Ric. Mat. 54, 591–595 (2005)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Leonetti, F., Petricca, P.V.: Existence for some vectorial elliptic problems with measure data. Riv. Math. Univ. Parma 5, 33–46 (2006)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Leonetti, F., Petricca, P.V.: Existence of bounded solutions to some nonlinear degenerate elliptic systems. Discrete Contin. Dyn. Syst, Ser B 11, 191–203 (2009)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Leonetti, F., Rocha, E., Staicu, V.: Quasilinear elliptic system with measure data. Nonlinear Anal. 154, 210–224 (2017)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Leonetti, F., Rocha, E., Staicu, V.: Smallness and cancellation in some elliptic systems with measure data. J. Math. Anal. Appl. 465, 885–902 (2018)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Mingione, G.: The Calderón-Zygmund theory for elliptic problems with measure data. Ann. Sc. Norm. Super. Pisa Cl. Sci. 6, 195–261 (2007)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Mingione, G.: Gradient estimates below the duality exponent. Math. Ann. 346, 571–627 (2010)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Mingione, G.: Nonlinear measure data problems. Milan J. Math. 79, 429–496 (2011)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Mingione, G.: La teoria di Calderon-Zygmund dal caso lineare a quello non lineare. Boll. Unione Mat. Ital. 9, 269–297 (2013)

    MathSciNet  Google Scholar 

  32. 32.

    Oppezzi, P., Rossi, A.M.: Esistenza di soluzioni per problemi unilateri con dato misura o in \(L^1\). Ric. Mat. 45, 491–513 (1996)

    Google Scholar 

  33. 33.

    Zhang, C., Zhou, S.: Entropy solutions for a non-uniformly parabolic equation. Manuscr. Math. 131, 335–354 (2010)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Zhou, S.: A note on nonlinear elliptic systems involving measures. Electron. J. Differ. Equ. 8, 1–6 (2000)

    MathSciNet  Google Scholar 

Download references


The authors would like to thank the anonymous referee for careful reading of the original version of this paper, and giving valuable comments and suggestions. Ren was partially supported by NSF of Hebei Province under grant no.A2018201285 and the research funding for high-level innovative talents of Hebei University under grant no.8012605. Gao was partially supported by NSF of Hebei Province under grant no. A2019201120.

Author information



Corresponding author

Correspondence to Wei Ren.

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

Gao, H., Huang, M., Deng, H. et al. Global integrability for solutions to quasilinear elliptic systems. manuscripta math. 164, 23–37 (2021).

Download citation

Mathematics Subject Classification

  • 35J57