Elliptic boundary value problems involving measures: Existence, regularity, and multiplicity
The authors consider semilinear elliptic boundary value problems of the types \[ \begin{aligned} -\Delta u&=f(x,u,\nabla u) + \mu \quad \text{in}\quad \Omega, \\ u&=0 \quad \text{on} \quad \Gamma_0,\\ \partial_\nu u&=g(x,u) + \sigma_1 \quad \text{on}\quad \Gamma_1, \end{aligned} \] and \[ \begin{aligned} -\Delta u&=h(x,u) + \mu \quad \text{in}\quad \Omega, \\ u&=\sigma_0 \quad \text{on}\quad \Gamma_0, \\ \partial_\nu u&=\sigma_1 \quad \text{on}\quad \Gamma_1, \end{aligned} \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^n\) \((n\geq 2)\), \(\Gamma:=\Gamma_0\cup \Gamma_1\) denotes its boundary, \(\Gamma_0\) and \(\Gamma_1\) are open in \(\Gamma\) and disjoint. Moreover, \(\mu\), \(\sigma_0\), \(\sigma_1\) are bounded Radon measures, and \(f\), \(g\), \(h\) are continuous functions satisfying some growth restrictions. The authors show that those problems can be put in a functional-analytical framework, and using this fact they derive existence, regularity and multiplicity results by methods from nonlinear functional analysis.
semilinear second-order boundary value problems
scientific article; zbMATH DE number 1449033