An elliptic semilinear equation with source term and boundary measure data: the supercritical case

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 $\mathrm{\Omega }$ is a either a bounded smooth domain or $\mathbb{R}\mathrm{\_}{+}^N$, $q\text{textgreater}1$ and $\sigma \in {\mathfrak{M}}^{+}(\mathrm{\partial }\mathrm{\Omega })$ is a nonnegative Radon measure on $\mathrm{\partial }\mathrm{\Omega }$. One of the criteria we obtain is expressed in terms of some Bessel capacities on $\mathrm{\partial }\mathrm{\Omega }$. 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 $q\mathrm{\_}1,q\mathrm{\_}2\ge 0,q\mathrm{\_}1+q\mathrm{\_}2\text{textgreater}1,q\mathrm{\_}2\text{textless}2$, $\sigma\in \mathfrak{M}(\partial\Omega)$isaRadonmeasureon$\partial\Omega$.