In this paper, we establish the Caccioppoli estimates for the nonlinear differential equation $$ - \overline{D}\bigl( \vert Dv \vert ^{p-2}Dv\bigr) = \lambda \vert v \vert ^{p-2}v, \quad 1< p< \infty ,$$
−
D
‾
(
|
D
v
|
p
−
2
D
v
)
=
λ
|
v
|
p
−
2
v
,
1
<
p
<
∞
,
where D is the Dirac operator. Moreover, we obtain general weighted versions of the Caccioppoli-type inequalities for the Dirac operators.