Non-homogeneous Divergence Structure Inequalities

We consider the quasilinear differential inequality

6.1.1

$$ divA\left( {x,u,Du} \right) + B\left( {x,u,Du} \right) \geqslant 0in\Omega , $$

where Ω is a bounded domain in ℝ

n

, and

A

and

B

satisfy the generic assumptions of Section 3.1. Here we shall extend the validity of Theorems 3.2.1 and 3.2.2 to the case when (6.1.1) is inhomogeneous, that is, there are constants

a

2

,

b

1

,

b

2

,

a

,

b

≥ 0 such that for all (

x, z, ξ

)

∈

Ω × ℝ

+

× ℝ

n

there holds, for

p

> 1,

6.1.2

$$ \begin{gathered} \left\langle {A\left( {x,z,\xi } \right),\xi } \right\rangle \geqslant \left| \xi \right|^p - a_2 z^p , \hfill \\ B\left( {x,z,\xi } \right) \leqslant b_1 \left| \xi \right|^{p - 1} + b_2 z^{p - 1} + b^{p - 1} , \hfill \\ \end{gathered} $$

while for

p

= 1,

6.1.3

$$ \left\langle {A\left( {x,z,\xi } \right),\xi } \right\rangle \geqslant \left| \xi \right| - a_2 z - a,B\left( {x,z,\xi } \right) \leqslant b $$

(in (6.1.3) we write

b

for

b

2

and discard the terms

b

1

|ξ|

p

−1

,

b

p

−1

). As in Section 3.1 the domain Ω is assumed to be bounded. This condition can be removed if Ω has finite measure and the boundary condition for |

x

| → ∞ is taken in the form (3.2.12).