  We introduce a generalized notion of semilinear elliptic partial differential
equations where the corresponding second order partial differential operator
$L$ has a generalized drift. We investigate existence and uniqueness of
generalized solutions of class $C^1$. The generator $L$ is associated with a
Markov process $X$ which is the solution of a stochastic differential equation
with distributional drift. If the semilinear PDE admits boundary conditions,
its solution is naturally associated with a backward stochastic differential
equation (BSDE) with random terminal time, where the forward process is $X$.
Since $X$ is a weak solution of the forward SDE, the BSDE appears naturally to
be driven by a martingale. In the paper we also discuss the uniqueness of a
BSDE with random terminal time when the driving process is a general
c{\`a}dl{\`a}g martingale.
