  We give new criteria for the existence of weak solutions to an equation with
a super linear source term \begin{align*}-\Delta u = u^q
~~\text{in}~\Omega,~~u=\sigma~~\text{on }~\partial\Omega\end{align*}where
$\Omega$ is a either a bounded smooth domain or $\mathbb{R}\_+^{N}$,
$q\textgreater{}1$ and $\sigma\in \mathfrak{M}^+(\partial\Omega)$ is a
nonnegative Radon measure on $\partial\Omega$. One of the criteria we obtain is
expressed in terms of some Bessel capacities on $\partial\Omega$. We also give
a sufficient condition for the existence of weak solutions to equation with
source mixed terms. \begin{align*} -\Delta u = |u|^{q\_1-1}u|\nabla u|^{q\_2}
~~\text{in}~\Omega,~~u=\sigma~~\text{on }~\partial\Omega \end{align*} where
$q\_1,q\_2\geq 0, q\_1+q\_2\textgreater{}1, q\_2\textless{}2$, $\sigma\in
\mathfrak{M}(\partial\Omega)$ is a Radon measure on $\partial\Omega$.
