On some quasilinear PDE's with singularities on the boundary. (Q1850186)

The author considers Dirichlet or initial value problems for the equations (inequalities) in \(\Omega\subset\mathbb R^n\) of types \(-\Delta_p u -\lambda a(x)u^{p-1} = b(x)u^\beta\delta^{-\alpha}(x)\), \(-\Delta_p u\geq b(x)u^\beta\delta^{-\alpha}(x)\), \(\frac{\partial^j u}{\partial t^j}-\Delta u\geq b(x)u^\beta\delta^{-\alpha}(x)\), where \(\delta(x) = \text{dist\,}(x, \partial\Omega)\), \(j=1, 2, \ldots\). He proves some existence results (there are at least one or two nontrivial positive solutions, or a countable family of nonpositive ones) or nonexistence results. The basic tools are Pohozaev fibering method and Mitidieri--Pohozaev nonlinear capacity method.