Singularly perturbed quadratically nonlinear Dirichlet problems

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