A class of quasilinear degenerate elliptic problems

We establish the existence of solutions for a class of quasilinear degenerate elliptic equations. The equations in this class satisfy a structure condition which provides ellipticity in the interior of the domain, and degeneracy only on the boundary. Equations of transonic gas dynamics, for example, satisfy this property in the region of subsonic flow and are degenerate across the sonic surface. We prove that the solution is smooth in the interior of the domain but may exhibit singular behavior at the degenerate boundary. The maximal rate of blow-up at the degenerate boundary is bounded by the “degree of degeneracy” in the principal coefficients of the quasilinear elliptic operator. Our methods and results apply to the problems recently studied by several authors which include the unsteady transonic small disturbance equation, the pressure-gradient equations of the compressible Euler equations, and the singular quasilinear anisotropic elliptic problems, and extend to the class of equations which satisfy the structure condition, such as the shallow water equation, compressible isentropic two-dimensional Euler equations, and general two-dimensional nonlinear wave equations. Our study provides a general framework to analyze degenerate elliptic problems arising in the self-similar reduction of a broad class of two-dimensional Cauchy problems.