*Published Paper*

**Inserted:** 22 jul 2018

**Last Updated:** 22 jul 2018

**Journal:** Journal of Functional Analysis

**Year:** 2015

**Doi:** https://www.sciencedirect.com/science/article/pii/S002212361500261X

**Abstract:**

We give new criteria for the existence of weak solutions to equation with source term
$
-\Delta u = u^q $ in
$\Omega$, $u=\sigma$ on $\partial\Omega $ where $q>1$, $\Omega$ is a either a bounded smooth domain or $\mathbb{R}_+^{N}$ and $\sigma\in \mathbf{M}^+(\partial\Omega)$ is a nonnegative Radon measure on $\partial\Omega$. In particular, one of the criteria 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 term.
$ -\Delta u =

u

^{q_1-1}u

\nabla u

^{q_2}$ in
$\Omega$, $u=\sigma$ on $\partial\Omega$ where $q_1,q_2\geq 0, q_1+q_2>1, q_2<2$, $\sigma\in \mathbf{M}(\partial\Omega)$ is a Radon measure on $\partial\Omega$.

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