In this paper we develop a new theory for the existence, localization and multiplicity of positive solutions for a class of non-variational, quasilinear, elliptic systems. In order to do this, we provide a fairly general abstract framework for the existence of fixed points of nonlinear operators acting on cones that satisfy an inequality of Harnack type. Our methodology relies on fixed point index theory. We also provide a non-existence result and an example to illustrate the theory.
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
weak Harnack inequality; fixed point index;p-Laplace operator; quasilinear elliptic system; positive weak solution; cone; multiplicity; nonexistence
G. Infante, M. Maciejewski, R. Precup, A topological approach to the existence and multiplicity of positive solutions of (p,q)-Laplacian systems, Dyn. Partial Differ. Equ. 12 (2015), no.3, 193-215, http://dx.doi.org/10.4310/DPDE.2015.v12.n3.a1
Dynamics of Partial Differential Equations
google scholar link
 A. Aghajani, J. Shamshiri, Multiplicity of positive solutions for quasilinear elliptic p-Laplacian systems, Electron. J. Differential Equations No. 111 (2012), 1–16.
 C. Azizieh, Ph. Clement. A priori estimates and continuation methods for positive solutions of p-Laplace equations, J. Differential Equations 179 (2002), 213-245.
 C. Azizieh, Ph. Clement, E. Mitidieri, Existence and a priori estimates for positive solutions of p-Laplace systems, J. Differential Equations 184 (2002), 422–442.
 Ph. Clement, J. Fleckinger, E. Mitidieri, F. de Thelin, Existence of positive solutions for a nonvariational quasilinear elliptic system, J. Differential Equations 166 (2000), 455–477.
 Ph. Clement, M. Garcıa-Huidobro, I. Guerra, R. Manasevich, On regions of existence and nonexistence of solutions for a system of p-q-Laplacians, Asymptot. Anal. 48 (2006), 1–18.
 K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
 S. El Manouni, K. Perera, R. Shivaji, On singular quasi-monotone (p, q)-Laplacian systems, Proc. Roy. Soc. Edinburgh Sect. A 142 (1012), 585–594.
 J. Fleckinger, J.-P. Gossez, P. Takac, 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), 731–734.
 J. Giacomoni, J. Hernandez, A. Moussaoui, Quasilinear and singular systems: the cooperative case. In Nonlinear elliptic partial differential equations, volume 540 of Contemp. Math., pages 79–94. Amer. Math. Soc., Providence, RI, 2011.
 A. Granas, J. Dugundji, Fixed Point Theory, Springer, New York, 2003.
 D. D. Hai, R. Shivaji, An existence result on positive solutions for a class of p-Laplacian systems, Nonlinear Anal. 56 (2004), 1007–1010.
 D. D. Hai, H. Wang, Nontrivial solutions for p-Laplacian systems, J. Math. Anal. Appl. 330 (2007), 186–194.
 G. Infante, P. Pietramala, Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations, Nonlinear Anal. 71 (2009), 1301–1310.
 P. Jebelean, R. Precup, Solvability of p, q-Laplacian systems with potential boundary conditions, Appl. Anal. 89 (2010), 221–228.
 J. Jost, Partial Differential Equations, Springer-Verlag, New York, 2002.
 K. Q. Lan, Z. Zhang, Nonzero positive weak solutions of systems of p-Laplace equations, J. Math. Anal. Appl. 394 (2012), 581–591.
 P. Lindqvist, Notes on the p-Laplace equation, Report 102. University of Jyv¨askyl¨a Department of Mathematics and Statistics, University of Jyv¨askyl¨a, Jyv¨askyl¨a, 2006.
 L. Montoro, B. Sciunzi, M. Squassina, Symmetry results for nonvariational quasi-linear elliptic systems, Adv. Nonlinear Stud. 10 (2010), 939–955.
 R. Precup, Moser-Harnack inequality, Krasnosel’skii type fixed point theorems in cones and elliptic problems, Topol. Methods Nonlinear Anal. 40 (2012), 301–313.
 R. Precup, Abstract weak Harnack inequality, multiple fixed points and p-Laplace equations, J. Fixed Point Theory Appl. 12 (2012), 193–206.
 Y. Shen, J. Zhang, Multiplicity of positive solutions for a semilinear p-Laplacian system with Sobolev critical exponent, Nonlinear Anal. 74 (2011), 1019–1030.
 N. S. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic. Comm. Pure Appl. Math. 20 (1967), 721–747.
 H. Wang, Existence and nonexistence of positive radial solutions for quasilinear systems, Discrete Contin. Dyn. Syst. suppl. (2009), 810–817.
 J. R. L. Webb, K. Q. Lan, Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type, Topol. Methods Nonlinear Anal. 27 (2006), 91–115.
 L. Wei, Z. Feng, Existence and nonexistence of solutions for quasilinear elliptic systems, Dyn. Partial Differ. Equ. 10 (2013), 25–42.