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We study the generalized boundary value problem for nonnegative solutions of of −∆u + g(u) = 0 in a bounded Lipschitz domain Ω, when g is continuous and nondecreasing. Using the harmonic measure of Ω, we define a trace in the class of outer regular Borel measures. We amphasize the case where g(u) = |u| q−1 u, q > 1. When Ω is (locally) a cone with vertex y, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that Ω possesses a tangent cone at every boundary point and q is subcritical, we prove an existence and uniqueness result for positive solutions with arbitrary boundary trace.1991 Mathematics Subject Classification. 35K60; 31A20; 31C15; 44A25; 46E35.