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We introduce a generalized notion of semilinear elliptic partial differentialequations where the corresponding second order partial differential operator$L$ has a generalized drift. We investigate existence and uniqueness ofgeneralized solutions of class $C^1$. The generator $L$ is associated with aMarkov process $X$ which is the solution of a stochastic differential equationwith distributional drift. If the semilinear PDE admits boundary conditions,its solution is naturally associated with a backward stochastic differentialequation (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 tobe driven by a martingale. In the paper we also discuss the uniqueness of aBSDE with random terminal time when the driving process is a generalc{\`a}dl{\`a}g martingale.

Elliptic PDEs with distributional drift and backward SDEs driven by a c{\`a}dl{\`a}g martingale with random terminal time