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We investigate the properties of certain elliptic systems leading, a~priori,to solutions that belong to the space of Radon measures. We show that if theproblem is equipped with a so-called asymptotic Uhlenbeck structure, then thesolution can in fact be understood as a standard weak solution, with oneproviso: analogously as in the case of minimal surface equations, theattainment of the boundary value is penalized by a measure supported on (asubset of) the boundary, which, for the class of problems under considerationhere, is the part of the boundary where a Neumann boundary condition isimposed.

On the existence of integrable solutions to nonlinear elliptic systems and variational problems with linear growth