### Moser-Harnack inequality, Krasnosel'skii type fixed point theorems in cones and elliptic problems

#### Abstract

Fixed point theorems of Krasnosel'skiĭ type are obtained for the localization

of positive solutions in a set defined by means of the norm and of

a semi-norm. In applications to elliptic boundary value problems, the

semi-norm comes from the Moser-Harnack inequality for nonnegative

superharmonic functions whose use is crucial for the estimations from below.

The paper complements and gives a fixed point alternative approach to our

similar results recently established in the frame of critical point theory.

It also provides a new method for discussing the existence and multiplicity

of positive solutions to elliptic boundary value problems.

of positive solutions in a set defined by means of the norm and of

a semi-norm. In applications to elliptic boundary value problems, the

semi-norm comes from the Moser-Harnack inequality for nonnegative

superharmonic functions whose use is crucial for the estimations from below.

The paper complements and gives a fixed point alternative approach to our

similar results recently established in the frame of critical point theory.

It also provides a new method for discussing the existence and multiplicity

of positive solutions to elliptic boundary value problems.

#### Keywords

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

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