We consider solutions to the nonlinear eigenvalue problem
\[
(
∗
)
A
(
x
,
u
→
)
u
→
+
λ
f
(
x
,
u
→
)
=
0
in
Ω
,
u
→
=
0
on
∂
Ω
,
u
→
=
0
,
on
∂
Ω
,
u
→
=
0
,
(*)\quad A(x,\vec u)\vec u + \lambda f(x,\vec u) = 0\:\quad {\text {in}}\,\Omega ,\quad \vec u = 0\:\quad {\text {on}}\,\partial \Omega ,\quad \vec u{\text { = }}0,\quad {\text {on}}\partial \Omega ,\quad \vec {u} = 0,
\]
where (*) is a quasilinear strongly coupled second order elliptic system of partial differential equations and
Ω
⊆
R
n
\Omega \subseteq \mathbf {R}^{n}
is a smooth bounded domain. We obtain lower bounds for
λ
\lambda
in the case where
f
(
x
,
u
→
)
f(x,\vec u)
has linear growth, and relations between
λ
,
Ω
\lambda ,\Omega
, and ess sup
|
u
→
|
|\vec u|
when
f
(
x
,
u
→
)
f(x,\vec u)
has sub- or superlinear growth. The estimates are based on integration by parts and application of certain Sobolev inequalities. We briefly discuss extensions to higher order systems.