Trace inequalities of the Sobolev type and nonlinear Dirichlet problems

Abstract

We discuss the solvability of Dirichlet problems of the type −∆p,wu = σ in Ω; u = 0 on ∂Ω, where Ω is a bounded domain in R n , ∆p,w is a weighted (p, w)-Laplacian and σ is a nonnegative locally finite Radon measure on Ω. We do not assume the finiteness of σ(Ω). We revisit this problem from a potential theoretic perspective and provide criteria for the existence of solutions by L p (w)-L q (σ) trace inequalities or capacitary conditions. Additionally, we apply the method to the singular elliptic problem −∆p,wu = σu −γ in Ω; u = 0 on ∂Ω and derive connection with the trace inequalities.

On a Class of Weighted p-Laplace Equation with Singular Nonlinearity

“…Further, these class of weights has been generalized to a class of p-admissible weights for the weighted p-Laplace equation (1.6). For such weights, nonexistence results has been discussed in Garain-Kinnunen [28] and existence results also established in Hara [33] for the purely singular nonlinearity g. The weighted anisotropic case is recently discussed in Bal-Garain [5,27] for a class of p-admissible weights.…”