Liouville comparison principles for solutions of semilinear elliptic second-order partial differential inequalities

Abstract

The purpose of this work is to obtain new Liouville comparison principles for entire weak solutions of semilinear elliptic second-order partial differential inequalities of the forms

and

on ℝ ⁿ , where n ≥ 2, q > 0 and ℒ is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergence form, in terms of the behaviour of the coefficients of the operator ℒ at infinity in a function space directly associated with ℒ.

Keywords:

• elliptic
• semilinear
• Liouville theorem
• comparison principle

AMS Subject Classifications::

• 35J61
• 35B08
• 35B51
• 35B53
• 35R45

Acknowledgements

This research was financially supported by the Alexander von Humboldt Foundation (AvH). The author is very grateful to AvH for the opportunity to visit the Mathematical Institute of Köln University and to Professor B. Kawohl for his cordial hospitality during this visit. He is also grateful to Professor B. Kawohl for the fruitful discussions of the results obtained which have helped to improve the presentation.