On Positive Solutions of the Equation $\mathfrak Au+Vu=0$

Let $X_t$ be a path of the continuous Markov process in the domain $D$ with boundary $\Gamma$ in a metric space, $\tau$ is the moment of reaching $\Gamma $; $\mathfrak{A}$ is the extended infinitesimal operator of the process and $V$ is a continuous non-negative function at $D$. The theorem reads as follows.
Let $\Gamma$ be regular in the sense of (1), $X_t$ be a strongly Feller process; then $\mathbf M_x\exp\left\{ \int_0^\tau{V(X_t )\,dt}\right\}<+\infty$ if and only if the equation (2) has a positive and continuous solution in $D\cup\Gamma$.
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 $\mathfrak{A}u+Vu$ ($u|_\Gamma=0$) by some global changes of domain is proved also.