Abstract weak Harnack inequality, multiple fixed points and p-Laplace equations


This paper presents an abstract theory for the existence, localization and multiplicity of fixed points in a cone. The key assumption is the property of the nonlinear operator of satisfying an inequality of Harnack type. In particular, the theory offers a completely new approach to the problem of positive solutions of quasilinear elliptic equations with p-Laplace operator.


Radu Precup
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania


Weak Harnack inequality; fixed point; fixed point index; p-Laplace operator; quasilinear elliptic equation.

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R. Precup, Abstract weak Harnack inequality, multiple fixed points and p-Laplace equations, J. Fixed Point Theory Appl. 12 (2012), 193-206, https://doi.org/10.1007/s11784-012-0091-2



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[1] C. Azizieh and Ph. Clement, A priori estimates and continuation methods for positive solutions of p-Laplace equations. J. Differential Equations 179 (2002), 213–245.
[2] K. Deimling, Nonlinear Functional Analysis. Springer, Berlin, 1985.
[3] G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian. Port. Math. 58 (2001), 339–378.
[4] P. Drabek and J. Hernandez, Existence and uniqueness of positive solutions for some quasilinear elliptic problems. Nonlinear Anal. 44 (2001), 189–204.
[5] L. H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations. Proc. Amer. Math. Soc. 120 (1994), 743–748.
[6] A. Granas and J. Dugundji, Fixed Point Theory. Springer, New York, 2003.
[7] D. D. Hai, On a class of sublinear quasilinear elliptic problems. Proc. Amer. Math. Soc. 131 (2003), 2409–2414.
[8] D. D. Hai and H. Wang, Nontrivial solutions for p-Laplacian systems. J. Math. Anal. Appl. 330 (2007), 186–194.
[9] J. Jost, Partial Differential Equations. Springer, New York, 2007.
[10] M. Kassmann, Harnack inequalities: An introduction. Bound. Value Probl. 2007, Article ID 81415, 21 pages, doi:10.1155/2007/81415.
[11] M. A. Krasnoselskii, Positive Solutions of Operator Equations. Noordhoff, Groningen, 1964.
[12] K. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities. J. Differential Equations 148 (1998), 407–421.
[13] P. Lindqvist, Notes on the p-Laplace Equation. Report/University of Jyvaskyla, Department of Mathematics and Statistics 102, 2006.
[14] M. Meehan and D. O’Regan, Positive Lp solutions of Hammerstein integral equations. Arch. Math. (Basel) 76 (2001), 366–376.
[15] J. Moser, On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14 (1961), 577–591.
[16] P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential. Comm. Partial Differential Equations 21 (1996), 721–733.
[17] P. Omari and F. Zanolin, An elliptic problem with arbitrarily small positive solutions. In: Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, FL, 1999), Electron. J. Differ. Equ. Conf. 5, Southwest Texas State Univ., San Marcos, TX, 2000, 301–308.
[18] D. O’Regan and R. Precup, Positive solutions of nonlinear systems with pLaplacian on finite and semi-infinite intervals. Positivity 11 (2007), 537–548.
[19] I. Peral, Multiplicity of solutions for the p-Laplacian. Preprint, Int. Center for Theoretical Physics, Trieste, 1997.
[20] R. Precup, Positive solutions of semi-linear elliptic problems via Krasnoselskii type theorems in cones and Harnack’s inequality. In: Mathematical Analysis
[21] R. Precup, Moser-Harnack inequality, Krasnoselskii type fixed point theorems in cones and elliptic problems. Topol. Methods Nonlinear Anal., to appear.
[22] R. Precup, Critical point theorems in cones and multiple positive solutions of elliptic problems. Nonlinear Anal. 75 (2012), 834–851.
[23] M. Ramaswamy and R. Shivaji, Multiple positive solutions for a class of pLaplacian equations. Differential Integral Equations 17 (2004), 1255–1261.
[24] S. H. Rasouli, G. A. Afrouzi and J. Vahidi, On positive weak solutions for some nonlinear elliptic boundary value problems involving the p-Laplacian. J. Math. Comp. Sci. 3 (2011), 94–101.
[25] D. Ruiz, A priori estimates and existence of positive solutions for strongly nonlinear problems. J. Differential Equations 199 (2004), 96–114.
[26] N. S. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations. Comm. Pure Appl. Math. 20 (1967), 721–747.
[27] H. Wang, On the existence of positive solutions for semilinear elliptic equations in the annulus. J. Differential Equations 109 (1994), 1–7.

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