Trace inequalities of the Sobolev type and nonlinear Dirichlet problems

We discuss the solvability of nonlinear Dirichlet problems of the type -Delta(p,w)u = sigma in Omega; u = 0 on a partial derivative Omega, where Omega is a bounded domain in 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 derivative Omega and derive connection with the trace inequalities.