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

Mateusz Maciejewski

DOI: http://dx.doi.org/10.12775/TMNA.2015.065

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 entails spectral bounds of the matrices associated with $f=(f_1,\ldots,f_m)$. We employ the degree theory from \cite{CwMac} for tangent perturbations of maximal monotone operators in Banach spaces.


Keywords


degree theory; tangency condition; cone; quasilinear elliptic system; positive weak solution

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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. Clement 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. Takac and F. de Thelin, Existence, nonexistence et principe de l'antimaximum pour le {$p$}-laplacien, C.R. Acad. Sci. Paris S'er. I Math. 321 (1995), no. 6, 731-734.

J. Fleckinger, R. Pardo and F. de Thelin, 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 ANRs, 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 div(vert nabla uvert sp {p-2}nabla u)+lambdavert uvert sp {p-2}u=0, Proc. Amer. Math. Soc. {109 (1990), no. 1, 157-164.

P. Lindqvist, Addendum: {O}n the equation div(vert nabla uvert sp {p-2}nabla u)+lambdavert uvert sp {p-2}u=0 [{P}roc. {A}mer. {M}ath. {S}oc. {109 (1990), no. 1, 157-164; {MR}1007505 (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.


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