On Positive Solutions of the Equation AU + Vu = 0

Let Xₜ be a path of the continuous Markov process in the domain D with boundary Γ in a metric space, τ is the moment of reaching Γ; A is the extended infinitesimal operator of the process and V is a continuous non-negative function at D. The theorem reads as follows. LetΓbe regular in the sense of (1), Xₜbe a strongly Feller process; thenMₓ exp \ ∫₀^τ V(Xₜ )dt \ < + ∞if and only if the equation (2) has a positive and continuous solution inD ∪ Γ. This theorem is applied to obtain different conditions, which guarantee the existence of the unique solution of the first boundary value problem for (2). Stability of the maximal eigenvalue of the operator Au + Vu(u|_Γ = 0) by some global changes of domain is proved also.