  With the terminal value $\xi^-$ admitting a certain exponential moment and
$\xi^+$ admitting every exponential moments or being bounded, we establish
several existence and uniqueness results for unbounded solutions of backward
stochastic differential equations (BSDEs) whose generator $g$ satisfies a
monotonicity condition with general growth in the first unknown variable $y$
and a convexity condition with quadratic growth in the second unknown variable
$z$. In particular, the generator $g$ may be not locally-Lipschitz continuous
in $y$. This generalizes some results reported in \cite{Delbaen 2011} by
relaxing the continuity and growth of $g$ in $y$. We also give an explicit
expression of the first process in the unique unbounded solution of a BSDE when
the generator $g$ is jointly convex in $(y,z)$ and has a linear growth in $y$
and a quadratic growth in $z$. Finally, we put forward the corresponding
comparison theorems for unbounded solutions of the preceding BSDEs. These
results are proved by those existing ideas and some innovative ones.
