Trace inequalities of the Sobolev type and nonlinear Dirichlet problems

We discuss the solvability of Dirichlet problems type $- \Delta_{p, w} u = \sigma$ in $\Omega$; $u 0$ on $\partial \Omega$, where $\Omega$ is a bounded domain $\mathbb{R}^{n}$, $\Delta_{p, w}$ weighted $(p, w)$-Laplacian and $\sigma$ nonnegative locally finite Radon measure $\Omega$. do not assume finiteness $\sigma(\Omega)$. revisit this problem from potential theoretic perspective provide criteria for existence solutions by $L^{p}(w)$-$L^{q}(\sigma)$ trace inequalities or capacitary conditions. Additionally, we apply method to singular elliptic \sigma u^{- \gamma}$ \Omega$ derive connection with inequalities.