Bounded and LpLp-weak solutions for nonlinear equations of measure-valued branching processes

We study the nonlinear Schrödinger type equation (Δ−c)u+c∑k=1∞qkuk=f, where cc is a bounded positive function. We may replace the Laplace operator with the generator of a Borel right Markov process with state space EE and consider a nonlinear term induced by a perturbing kernel. We solve this type of equation in an LpLp-weak sense, involving positive definite extensions of the solution from EE to the set of all finite positive measures on EE. The extensions to the set of measures are related to the classical connection (from the works of M. Nagasawa, N. Ikeda, S. Watanabe, and M.L. Silverstein) between this nonlinear equation and the discrete branching Markov processes. We also investigate the existence of the bounded solutions for the Dirichlet problem associated to the above equation, with bounded boundary data. Our approach is based on probabilistic and analytic potential theoretical methods, used on both spaces EE and the set of all finite configurations of EE, like the Revuz formula for the continuous additive functionals and the perturbation with kernels of the transition functions and resolvents.