Nonlinear Elliptic Equations for Measures

Nonlinear elliptic equations are studied for probability measures. It is supposed that a nonnegative Borel measure satisfies the weak elliptic equation, and it is found that for a fixed measure, this equation is linear. If measures are such that their densities locally uniformly converge to the density of a measure, then the functions converge on each ball. Since the measures are probability, it follows that the convergence of their densities implies convergence in variation. If the mappings are defined only for measures and satisfy certain conditions, then, in the class, the equation has a solution. The existence of a solution of the transport equation is ensured if the function k is continuous on the space of measures with the weak topology.