We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled bywhere f is an irregular datum, possibly a measure, and h is a continuous function that may blow up at zero. We also provide regularity results on both the solution and the lower order term depending on the regularity of the data, and we discuss their optimality.

Abstract. We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model iswhere Ω is an open bounded subset of R N . Here γ > 0, f is a nonnegative function on Ω, and µ is a nonnegative bounded Radon measure on Ω.

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the formis a continuous function which may become singular at s = 0 + , and f is a nonnegative datum in L N,∞ (Ω) with suitable small norm. Uniqueness of solutions is also shown provided h is decreasing and f > 0. As a by-product of our method a general theory for the same problem involving the p-laplacian as principal part, which is missed in the literature, is established. The main assumptions we use are also further discussed in order to show their optimality.

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