A Liouville comparison principle for solutions of semilinear elliptic partial differential inequalities

Abstract

This work is devoted to the study of a Liouville-type comparison principle for entire weak solutions of semilinear elliptic partial differential inequalities of the form

$$ \mathcal{L}u+|u|^{q-1}u \leq\mathcal{L}v+|v|^{q-1}v, $$

where $q>0$ is a given real number and $\mathcal{L}$ is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergence form given by the relation

$$ \mathcal{L}=\sum_{i,j=1}^n\frac{\partial}{\partial x_i}\bigg[a_{ij}(x)\frac{\partial}{\partial x_j}\bigg]. $$

We assume that $n\geq2$, that the coefficients $a_{ij}(x)$, $i,j=1,\dots,n$, are measurable bounded functions on $\mathbb{R}^n$ such that $a_{ij}(x)=a_{ji}(x)$ and that the corresponding quadratic form is non-negative. The results obtained in this work were announced by the author in 2005.