Topological Methods in Nonlinear Analysis

Positive solutions to $p$-Laplace reaction-diffusion systems with nonpositive right-hand side

Mateusz Maciejewski

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The aim of the paper is to show the existence of positive solutions to the elliptic system of partial differential equations involving the $p$-Laplace operator \[ \begin{cases} -\Delta_p u_i(x) = f_i(u_1 (x),u_2(x),\ldots,u_m(x)), & x\in \Omega,\ 1\leq i\leq m, \\ u_i(x)\geq 0, & x\in \Omega,\ 1\leq i\leq m,\\ u(x) = 0, & x\in \partial \Omega. \end{cases} \] We consider the case of nonpositive right-hand side $f_i$, $i=1,\ldots,m$. The sufficient conditions entail spectral bounds of the matrices associated with $f=(f_1,\ldots,f_m)$. We employ the degree theory from [A. Ćwiszewski and M. Maciejewski, Positive stationary solutions for $p$-Laplacian problems with nonpositive perturbation, J. Differential Equations 254 (2013), no. 3, 1120-1136] for tangent perturbations of maximal monotone operators in Banach spaces.

Article information

Topol. Methods Nonlinear Anal., Volume 46, Number 2 (2015), 731-754.

First available in Project Euclid: 21 March 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Maciejewski, Mateusz. Positive solutions to $p$-Laplace reaction-diffusion systems with nonpositive right-hand side. Topol. Methods Nonlinear Anal. 46 (2015), no. 2, 731--754. doi:10.12775/TMNA.2015.065.

Export citation


  • A. Aghajani and J. Shamshiri, Multiplicity of positive solutions for quasilinear elliptic $p$-Laplacian systems, Electron. J. Differential Equations 2012 (2012), no. 111, 1–16.
  • C. Azizieh, P. Clément and E. Mitidieri, Existence and a priori estimates for positive solutions of $p$-Laplace systems, J. Differential Equations 184 (2002), no. 2, 422–442.
  • F.H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247–262.
  • A. Ćwiszewski and W. Kryszewski, Constrained topological degree and positive solutions of fully nonlinear boundary value problems, J. Differential Equations 247 (2009), no. 8, 2235–2269.
  • A. Ćwiszewski and M. Maciejewski, Positive stationary solutions for $p$-Laplacian problems with nonpositive perturbation, J. Differential Equations 254 (2013), no. 3, 1120–1136.
  • J. Fleckinger, J.-P. Gossez, P. Takáč and F. de Thélin, Existence, nonexistence et principe de l'antimaximum pour le $p$-laplacien, C.R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 6, 731–734.
  • J. Fleckinger, R. Pardo and F. de Thélin, Four-parameter bifurcation for a $p$-Laplacian system, Electron. J. Differential Equations 2001 (2001), no. 6, 1–15. (electronic), 2001.
  • F.R. Gantmacher, The Theory of Matrices, Vols. 1, 2, Chelsea Publishing Co., New York, 1959.
  • D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer–Verlag, Berlin, 1977; Grundlehren der Mathematischen Wissenschaften, Vol. 224.
  • A. Granas, The Leray–Schauder index and the fixed point theory for arbitrary \romANRs, Bull. Soc. Math. France 100 (1972), 209–228.
  • A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer–Verlag, New York, 2003.
  • D.D. Hai and H. Wang, Nontrivial solutions for $p$-Laplacian systems, J. Math. Anal. Appl. 330 (2007), no. 1, 186–194.
  • G. Infante, M. Maciejewski and R. Precup, A topological approach to the existence and multiplicity of positive solutions of $(p,q)$-Laplacian systems, preprint (, 2014.
  • W. Kryszewski and M. Maciejewski, Positive solutions to partial differential inclusions: degree-theoretic approach (in preparation).
  • K.Q. Lan and Z. Zhang, Nonzero positive weak solutions of systems of $p$-Laplace equations, J. Math. Anal. Appl. 394 (2012), no. 2, 581–591.
  • P. Lindqvist, On the equation ${\rm div}\,(\vert \nabla u\vert \sp {p-2}\nabla u)+\lambda\vert u\vert \sp {p-2}u=0$, Proc. Amer. Math. Soc. 109 (1990), no. 1, 157–164.
  • ––––, Addendum: On the equation ${\rm div}(\vert \nabla u\vert \sp {p-2}\nabla u)+\lambda\vert u\vert \sp {p-2}u=0$ [Proc. Amer. Math. Soc. 109 (1990), no. 1, 157–164; MR1007505 (90h:35088)], Proc. Amer. Math. Soc. 116 (1992), no. 2, 583–584.
  • Y. Shen and J. Zhang, Multiplicity of positive solutions for a semilinear $p$-Laplacian system with Sobolev critical exponent, Nonlinear Anal. 74 (2011), no. 4, 1019–1030.
  • H. Wang, Existence and nonexistence of positive radial solutions for quasilinear systems, Discrete Contin. Dyn. Syst., 2009, (Dynamical Systems, Differential Equations and Applications. 7th AIMS Conference, suppl.), 810–817.