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We study the existence of solutions of the nonlinear problem [GRAPHICS] where p is a bounded measure and g is a continuous nondecreasing function such that g(0) = 0. In this paper, we assume that the nonlinearity g satisfies [GRAPHICS] Problem (0.1) need not have a solution for every measure A. We prove that, given p, there exists a "closest" measure mu* for which (0.1) can be solved. We also explain how assumption (0.2) makes problem (0.1) different from the case where g(t) is defined for every t is an element of R

Elliptic equations with vertical asymptotes in the nonlinear term