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

In this work we analyze existence, nonexistence, multiplicity and regularity of solution to problem -Delta u = beta(u)vertical bar del u vertical bar(2) + lambda f(x) in Omega, u = 0 on partial derivative Omega where beta is a continuous nondecreasing positive function and f belongs to some suitable Lebesgue spaces. (c) 2005 Elsevier Inc. All rights reserved

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We prove the existence of solutions of nonlinear elliptic equations with firstorder terms having "natural growth" with respect to the gradient. The assumptions on the source terms lead to the existence of possibly unbounded solutions (though with exponential integrability). The domain Ω is allowed to have infinite Lebesgue measure.