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viscosity subsolution; viscosity supersolution; mean curvature equation; pseudo $p$-Laplace equation
In the present paper we study the Dirichlet boundary value problem for quasilinear elliptic equations including non-uniformly and degenerate ones. In particular, we consider mean curvature equation and pseudo p-Laplace equation as well. It is well-known that the proof of the existence of continuous viscosity solutions is based on Ishii’s implementation of Perron’s method. In order to use this method one has to produce suitable subsolution and supersolution. Here we introduce new methods to construct subsolutions and supersolutions for the above mentioned problems. Using these subsolutions and supersolutions one may prove the existence of unique continuous viscosity solution for a wide class of degenerate and non-uniformly elliptic equations.
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