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.