Moser-Harnack inequality, Krasnoselskii type fixed point theorems in cones and elliptic problems


The compression-expansion fixed point theorems of M. A. Krasnoselʹskiĭ have been extensively used as a basic tool for the existence and localization of positive solutions to ordinary differential equations, but almost never applied to partial differential equations, except particular situations which can be reduced to ordinary differential equations, such as the case of radial solutions.
In this paper, with the help of the Moser-Harnack inequality for nonnegative super-harmonic functions, the author determines a suitable cone of functions where the Krasnoselʹskiĭ technique works for semilinear elliptic equations. In effect, the author provides a new method to establish the existence and multiplicity of positive solutions for elliptic boundary value problems.


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


Fixed point index; cone; elliptic equation; positive solution; Moser–Harnack inequality.

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R. Precup, Moser-Harnack inequality, Krasnoselskii type fixed point theorems in cones and elliptic problems, Topol. Methods Nonlinear Anal. 40 (2012), 301-313.


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