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Inhomogeneous Dirichlet problems involving the infinity-Laplacian. (English) Zbl 1258.35094

The authors provide a self-contained account of the inhomogeneous Dirichlet problem \(\Delta_\infty u=f(x,u)\) where \(u\) assumes prescribed continuous data on the boundary of bounded domains. It is employed a combination of the Perron’s method and a priori estimates to give general sufficient conditions on the right-hand side \(f\) that would ensure existence of viscosity solutions to the Dirichlet problem. It is described a class of inhomogeneous terms for which the corresponding Dirichlet problem has no solution in any domain with large in-radius.

MSC:

35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35J70 Degenerate elliptic equations
35D40 Viscosity solutions to PDEs
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