  We study a backward stochastic differential equation whose terminal condition
is an integrable function of a local martingale and generator has bounded
growth in $z$. When the local martingale is a strict local martingale, the BSDE
admits at least two different solutions. Other than a solution whose first
component is of class D, there exists another solution whose first component is
not of class D and strictly dominates the class D solution. Both solutions are
$\mathbb{L}^p$ integrable for any $0<p<1$. These two different BSDE solutions
generate different viscosity solutions to the associated quasi-linear partial
differential equation. On the contrary, when a Lyapunov function exists, the
local martingale is a martingale and the quasi-linear equation admits a unique
viscosity solution of at most linear growth.
