On the behaviour of entire solutions of semilinear elliptic second-order partial differential inequalities

ABSTRACT

We study the behaviour of solutions $(u,v)$ of semilinear partial differential inequalities of the form $-Lu+|u{|}^{q-1}u\le -Lv+|v{|}^{q-1}v,$ which are defined, measurable and locally bounded in the whole space ${\mathbb{R}}^n$ and which belong locally to a Sobolev-type function space associated with a linear partial differential operator $L=\sum _{i,j=1}^n\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\left(a_{ij}\left(x\right)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\right)$ defined in ${\mathbb{R}}^n$, where $n\ge 2$ and q>1. We assume that the coefficients $a_{ij}\left(x\right)$ of the operator L are measurable, locally bounded and such that $a_{ij}\left(x\right)=a_{ji}\left(x\right)$, and that the quadratic form associated with the operator L is non-negative definite.

COMMUNICATED BY:

• D. Repovš

KEYWORDS:

• Capacity
• entire solution
• semilinear elliptic partial differential inequality

AMS SUBJECT CLASSIFICATIONS:

• 35J91 (Primary)
• 35B08
• 35B40
• 35J70
• 35R45 (Secondary)

Disclosure statement

No potential conflict of interest was reported by the author.