Trace inequalities of the Sobolev type and nonlinear Dirichlet problems

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