Regularizing effect of the interplay between coefficients in some elliptic equations

We study the regularizing effect of the interaction between the coefficient of the zero order term and the datum in some nonlinear Dirichlet problems.The simplest example is the linear problem{∫ΩM(x)∇u∇φ+∫Ωa(x)uφ=∫Ωf(x)φ,∀φ∈W01,2(Ω)∩L∞(Ω), where Ω   is a bounded open set of RNRN, M   is a bounded elliptic matrix and 0≤a(x)∈L1(Ω)0≤a(x)∈L1(Ω). Even if f(x)f(x) only belongs to L1(Ω)L1(Ω), the assumptionthere existsQ>0such that|f(x)|≤Qa(x) implies the existence of a weak solution u   belonging to W01,2(Ω) and to L∞(Ω)L∞(Ω).