On classes of well-posedness for quasilinear diffusion equations in the whole space

Well-posedness classes for degenerate elliptic problems in RN under the form u = ∆ϕ(x, u) + f(x), with locally (in u) uniformly continuous nonlinearities, are explored. While we are particularly interested in the L∞ setting, we also investigate about solutions in L1loc and in weighted L1 spaces. We give some sufficient conditions in order that the uniqueness and comparison properties hold for the associated solutions; these conditions are expressed in terms of the moduli of continuity of u 7→ ϕ(x, u). Under additional restrictions on the dependency of ϕ on x, we deduce the existence results for the corresponding classes of solutions and data. Moreover, continuous dependence results follow readily from the existence claim and the comparison property. In particular, we show that for a general continuous non-decreasing nonlinearity ϕ: R 7→ R, the space L∞ (endowed with the L1loc topology) is a well-posedness class for the problem u = ∆ϕ(u) + f(x). © 2021 American Institute of Mathematical Sciences. All rights reserved.