Existence of bounded solutions for semilinear degenerate elliptic equations with absorption term (Q1269351)

Let \(N\geq 1\) and \(p>1\). Let \(F\) be a compact set and \(\Omega\) be a bounded open set of \(\mathbb{R}^N\) satisfying \(F\subset\Omega\subset\mathbb{R}^N\). We also set \(\Omega'=\Omega\setminus\partial F\). Define \(P=-\text{div} (A(x)\nabla \cdot)\), where \(A(x)\in C^1(\Omega')\) is positive in \(\Omega\setminus F\) and vanishes in \(\overset\circ F\). First we consider removable singularities of solutions for degenerate semilinear elliptic equations \[ Pu+B(x) Q(u)=f(x), \quad\text{in }\Omega'. \] Here \(Q(u)\) is a nonlinear term. Then we show the existence of a bounded solution in \(\Omega\) which coincides with \(u\) on \(\Omega'=\Omega\setminus\partial F\).