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