Trace inequalities of the Sobolev type and nonlinear Dirichlet problems

Abstract

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.

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Cited by in corpus (2)

• Strong barriers for weighted quasilinear equations
• A stability result for elliptic equations with singular nonlinearity and its applications to homogenization problems