On a Liouville-type comparison principle for solutions of semilinear elliptic partial differential inequalities

Abstract

This Note is devoted to the study of a Liouville-type comparison principle for entire weak solutions of semilinear elliptic partial differential inequalities of the form L u + | u | q − 1 u ⩽ L v + | v | q − 1 v , where q > 0 is a given number and L is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergent form given formally by the relation L = ∑ i , j = 1 n ∂ ∂ x i [ a i j ( x ) ∂ ∂ x j ] . We assume that n ⩾ 2 , that the coefficients a i j ( x ) , i , j = 1 , … , n , are measurable bounded functions on R n such that a i j ( x ) = a j i ( x ) , and that the corresponding quadratic form is non-negative. The results obtained in this work complete similar results on solutions of quasilinear elliptic partial differential inequalities announced in Kurta [C. R. Acad. Sci. Paris, Ser. I 336 (11) (2003) 897–900]. To cite this article: V.V. Kurta, C. R. Acad. Sci. Paris, Ser. I 341 (2005).