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Transactions of the American Mathematical Society

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Singularly perturbed quadratically nonlinear Dirichlet problems

Author: Albert J. DeSanti
Journal: Trans. Amer. Math. Soc. 298 (1986), 733-746
MSC: Primary 35B25; Secondary 35J65
MathSciNet review: 860390
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Abstract: 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.

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