We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with Singular lower order terms that have natural growth with respect to the gradient, whose model is {-Delta u + vertical bar del u vertical bar(2)/u(gamma) = f in Omega. u = 0 on partial derivative Omega. where Omega is an open bounded subset of R, gamma > 0 and f is a function which is strictly positive on every compactly contained subset of Omega. As a consequence of our main results, we prove that the condition gamma < 2 is necessary and sufficient for the existence of solutions in H(0)(1) (Omega) for every sufficiently regular f as above. (C) 2009 Elsevier Inc. All rights reserved

Abstract. Let Ω ⊆ R N a bounded open set, N ≥ 2, and let p > 1; we prove existence of a renormalized solution for parabolic problems whose model isin Ω,where T > 0 is any positive constant, µ ∈ M (Q) is a any measure with bounded variation over Q = (0, T ) × Ω, and u 0 ∈ L 1 (Ω), and −∆pu = −div(|∇u| p−2 ∇u) is the usual p-laplacian.

Abstract. Given a parabolic cylinder Q = (0, T ) × , where ⊂ R N is a bounded domain, we prove new properties of solutions ofwith Dirichlet boundary conditions, where µ is a finite Radon measure in Q. We first prove a priori estimates on the p-parabolic capacity of level sets of u. We then show that diffuse measures (i.e., measures which do not charge sets of zero parabolic p-capacity) can be strongly approximated by the measures, and we introduce a new notion of renormalized solution based on this property. We finally apply our new approach to prove the existence of solutions offor any function h such that h(s)s ≥ 0 and for any diffuse measure µ; when h is nondecreasing, we also prove uniqueness in the renormalized formulation. Extensions are given to the case of more general nonlinear operators in divergence form.

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.

Abstract. We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems likewith vanishing conditions at infinity. Here µ is a bounded Radon measure whose support is compactly contained in R N , N ≥ 2, and −(∆) s is the fractional Laplace operator of order s ∈ (1/2, 1).

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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