Remark on closedness of perturbation set and regularity of associated extremal solution (Q939430)

The Dirichlet problem for nonlinear elliptic equation \(-\Delta u=f(u)+g(x)\) in bounded smooth domain \(\Omega\) in \(\mathbb{R}^N, N\geq 2\) is considered in the paper. Let \(K\) be the set of \(g\in C^\alpha (\Omega)\) such that the problem has a classical solution and \(\overline{u}\) be the minimal solution. The closeness of \(K\) and continuity of the mapping \(g\rightarrow \overline{u}(g)\) are discussed.