Uniformly elliptic equations with concave growth in the gradient and measures

Abstract

We deal with quasilinear elliptic problems with measure data:Lw = H(x, w, ∇w) + µ in Ωw = 0 on ∂Ω,(0.1)where Lw := −div(A(x)∇w) with A = A(x) a bounded, coercive, and symmetric matrix field, theHamiltonian H has at most q-growth in the gradient for 0 < q < 1, and µ is any Radon measure.We employ the compactness of the Green operator associated to L from the space of measures toW1,p0(Ω) for all p ∈ [1, N/(N − 1)) together with fixed point arguments to solve problem (0.1) forany measure µ. Moreover, we provide explicit estimates of the solution in terms of the data. As anapplication, stability results are given. We also give conditions for the existence of W1,20-solutionsthrough the classic theory of monotone and coercive operators. In any case, we do not impose anysize restriction on µ and any sign condition on H.