Viscosity Subsolutions and Supersolutions for Non-uniformly and Degenerate Elliptic Equations

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. 0. Introduction and main results Consider the following problem − n ∑ i,j=1 aij(x,∇u)uxixj + f(x, u,∇u) = 0 in Ω ⊂ Rn , (0.1) u(x) = 0 on ∂Ω , (0.2) where for x ∈ Ω, p ∈ R and ξ = (ξ1, . . . , ξn) ∈ R (0.3) n ∑ i,j=1 aij(x,p)ξiξj ≥ 0 . In this paper we focus our attention on the problem of construction of viscosity subsolutions and supersolutions for equation (0.1) that satisfy boundary condition (0.2). Put Φ(x, u,∇u,∇2u) = − n ∑ i,j=1 aij(x,∇u)uxixj + f(x, u,∇u) , where we suppose that Φ(x, u,∇u,∇2u) is a continuous function in all its variables. Let us recall the definition of continuous viscosity suband supersolutions of (0.1). According to [6], u is a continuous viscosity subsolution of Φ = 0 if u is upper 2000 Mathematics Subject Classification: primary 35J60; secondary 35D05.