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We prove existence, regularity and nonexistence results for problems whose model is -Lambda u = f(x)/u gamma in Omega, with zero Dirichlet conditions on the boundary of an open, bounded subset Omega of R(N). Here gamma > 0 and f is a nonnegative function on Omega. Our results will depend on the summability of f in some Lebesgue spaces, and on the values of gamma (which can be equal, larger or smaller than 1)

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In this paper we give summability results for the gradients of solutions of nonlinear parabolic equations whose model iswith homogeneous Cauchy Dirichlet boundary conditions, where p>1 and + is a bounded measure on 0_(0, T ). We also study how the summability of the gradient improves if the measure + is a function in L m (0_(0, T )), with m``small.'' Moreover we give a new proof of the existence of a solution for problem (P). Academic Press

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We show existence and regularity of solutions in R N to nonlinear elliptic equations of the form −div A(x, Du) + g(x, u) = f , when f is just a locally integrable function, under appropriate growth conditions on A and g but not on f. Roughly speaking, in the model case −∆ p (u) + |u| s−1 u = f , with p > 2 − (1/N), existence of a nonnegative solution in R N is guaranteed for every nonnegative f ∈ L 1 loc (R N) if and only if s > p − 1.