The effects of convolution and gradient dependence on a parametric Dirichlet problem

  • Dumitru MotreanuEmail author
  • Calogero Vetro
  • Francesca Vetro
Original Paper
Part of the following topical collections:
  1. Theory of PDEs


Our objective is to study a new type of Dirichlet boundary value problem consisting of a system of equations with parameters, where the reaction terms depend on both the solution and its gradient (i.e., they are convection terms) and incorporate the effects of convolutions. We present results on existence, uniqueness and dependence of solutions with respect to the parameters involving convolutions.


Dirichlet problem Convolution System of elliptic equations \((p{, } q)\)-Laplacian Parametric problems 

Mathematics Subject Classification

35J45 35J55 


  1. 1.
    Adomian, G., Rach, R.: On the solution of nonlinear differential equations with convolution product. J. Math. Anal. Appl. 114(1), 171–175 (1986)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer, New York (2011)zbMATHGoogle Scholar
  3. 3.
    Carl, S., Le, V.K., Motreanu, D.: Nonsmooth variational problems and their inequalities. Comparison principles and applications. Springer, New York (2007)CrossRefGoogle Scholar
  4. 4.
    Carl, S., Motreanu, D.: Extremal solutions for nonvariational quasilinear elliptic systems via expanding trapping regions. Monatsh. Math. 182(4), 801–821 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    De Figueiredo, D., Girardi, M., Matzeu, M.: Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques. Differ. Integr. Equ. 17(1–2), 119–126 (2004)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Faraci, F., Motreanu, D., Puglisi, D.: Positive solutions of quasi-linear elliptic equations with dependence on the gradient. Calc. Var. Part. Differ. Equ. 54(1), 525–538 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Faria, L.F.O., Miyagaki, O.H., Motreanu, D.: Comparison and positive solutions for problems with \((p, q)\)-Laplacian and convection term. Proc. Edinb. Math. Soc. 57, 687–698 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Faria, L.F.O., Miyagaki, O.H., Motreanu, D., Tanaka, M.: Existence results for nonlinear elliptic equations with Leray-Lions operator and dependence on the gradient. Nonlinear Anal. 96, 154–166 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Le Dret, H.: Nonlinear elliptic partial differential equations. An introduction. Translated from the 2013 French Edition. Universitext, Springer, Cham (2018)Google Scholar
  10. 10.
    Mishra, N., Moundekar, M., Khalode, M., Shrivastava, M.K.: To implement convolution in image processing. Int. J. Sci. Eng. Technol. Res. (LISETR) 5(2), 604–606 (2016)Google Scholar
  11. 11.
    Motreanu, D., Motreanu, V.V.: Non-variational elliptic equations involving \((p,q)\)-Laplacian, convection and convolution, Pure Appl. Funct. Anal. (in print)Google Scholar
  12. 12.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)CrossRefGoogle Scholar
  13. 13.
    Motreanu, D., Vetro, C., Vetro, F.: A parametric Dirichlet problem for systems of quasilinear elliptic equations with gradient dependence. Numer. Funct. Anal. Optim. 37, 1551–1561 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Motreanu, D., Vetro, C., Vetro, F.: Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method. Discrete Contin. Dyn. Syst. Ser. S 11(2), 309–321 (2018)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Papageorgiou, N.S., Vetro, C.: Superlinear \((p(z), q(z))\)-equations. Complex Var. Elliptic Equ. 64(1), 8–25 (2019)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Papageorgiou, N.S., Vetro, C., Vetro, F.: Multiple solutions for \((p,2)\)-equations at resonance. Discrete Contin. Dyn. Syst. Ser. S 12(2), 347–374 (2019)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Papageorgiou, N.S., Vetro, C., Vetro, F.: \((p,2)\)-equations resonant at any variational eigenvalue. Complex Var. Ellipt. Equ. (2018). CrossRefzbMATHGoogle Scholar
  18. 18.
    Papageorgiou, N.S., Vetro, C., Vetro, F.: \((p,2)\)-equations with a crossing nonlinearity and concave terms. Appl. Math. Optim. (2018). CrossRefzbMATHGoogle Scholar
  19. 19.
    Ruiz, D.: A priori estimates and existence of positive solutions for strongly nonlinear problems. J. Differ. Equ. 199(1), 96–114 (2004)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Tanaka, M.: Existence of a positive solution for quasilinear elliptic equations with a nonlinearity including the gradient. Bound. Value Probl. 2013(173), 1–11 (2013)MathSciNetGoogle Scholar
  21. 21.
    Zeidler, E.: Nonlinear Functional Analysis and Its Applications, vol. II B. Springer, Berlin (1990)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PerpignanPerpignanFrance
  2. 2.College of ScienceYulin Normal UniversityYulinPeople’s Republic of China
  3. 3.Department of Mathematics and Computer ScienceUniversity of PalermoPalermoItaly
  4. 4.Nonlinear Analysis Research GroupTon Duc Thang UniversityHo Chi Minh CityVietnam
  5. 5.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam

Personalised recommendations