The author studies an elliptic Dirichlet problem
with and the space of Radon measures and an open bounded set of with . For there was a unique variational solution in . For one did not find solutions in and some weaker formulations had been introduced. Solutions were given by Stampacchia or by Boccardo and Gallouët by an approximation of . The author shows equivalences to other weaker formulations to see that the one of Boccardo and Gallouët is weaker than the one used by Stampacchia and to see that the solutions of Boccardo and Gallouët are solutions of Stampacchia. Because the Stampacchia-formulation ensures the uniqueness of , the approximative solution of Boccardo and Gallouët is unique.
Uniqueness for the Boccardo-Gallouët-formulation is not ensured and for and the unit ball of the author gives an adaption of the counterexample of Serrin for getting a nontrivial solution for such that for all and but with trace in which gives two solutions in this space.