The Dirichlet problem for singularly perturbed elliptic equations of the form
ε
Δ
u
=
A
(
x
,
u
)
∇
u
⋅
∇
u
+
B
(
x
,
u
)
⋅
∇
u
+
C
(
x
,
u
)
\varepsilon \Delta u = A({\mathbf {x}},u)\nabla u \cdot \nabla u + {\mathbf {B}}({\mathbf {x}},u) \cdot \nabla u + C({\mathbf {x}},u)
in
Ω
∈
E
n
\Omega \in {E^n}
is studied. Under explicit and easily checked conditions, solutions are shown to exist for
ε
\varepsilon
sufficiently small and to exhibit specified asymptotic behavior as
ε
→
0
\varepsilon \to 0
. The results are obtained using a method based on the theory of partial differential inequalities.