Algebraic investigation of a nonlinear and singular Dirichlet problem (Q1974097)

Let \(\Omega\) be a bounded open subset of \(\mathbb{R}^d\) with boundary of class \(C^\infty\) and suppose \(\theta\in C^\infty(\overline\Omega\times \mathbb{R})\) satisfies certain growth conditions. The authors first solve the regular Dirichlet problem \[ -\Delta u(x)+ \theta(x, u(x))= f(x), \] when \(x\in\Omega\), \(u(x)= g(x)\) when \(x\in\partial\Omega\) for a unique \(u\in C^\infty(\overline\Omega)\), where \(f\in C^\infty(\overline\Omega)\) and \(g\in C^\infty(\partial\Omega)\) are given. Their main result develops an idea of \textit{A. Delcroix} and \textit{D. Scarpalezos} [Integral Transforms, Spec. Funct. 6, No. 1-4, 181-190 (1998; Zbl 0919.46027)] in order to solve the Dirichlet problem when \(f\), \(g\), and \(u\) belong to a class of generalized functions derived from the algebraic structure of a quotient ring and the topological structure of a semi-normed algebra.