Existence results for some elliptic equations with unbounded coefficients
Luigi Orsina · Asymptotic Analysis · 2003
where Ω is a bounded open subset of RN , N 2, and μ is a bounded Radon measure on Ω. The main feature of this problem is the fact that the elliptic differential operator is not well defined on the whole of H1 0 (Ω) (which can be thought as the “natural” space to deal with this problem), since the function |∇u|/(1 − |u|) may not be in L1(Ω) if u takes the values ±1. It is nevertheless possible to define, for n in N, an approximating function hn as hn(s) = 1/(1−|s|) if 1/(1−|s|) n, and n otherwise, and solve the approximating problems { − div ( hn(un)∇un ) = μ in Ω, un = 0 on ∂Ω,