Topological Methods in Nonlinear Analysis

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

Mateusz Maciejewski

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Abstract

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

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

Dates
First available in Project Euclid: 21 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1458588659

Digital Object Identifier
doi:10.12775/TMNA.2015.065

Mathematical Reviews number (MathSciNet)
MR3494968

Zentralblatt MATH identifier
1381.35062

Citation

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. https://projecteuclid.org/euclid.tmna/1458588659


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References

  • 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 (http://arxiv.org/abs/1401.1355v2), 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.