Summary: We consider Dirichlet problems of the form \(-|x|^\alpha \Delta u=\lambda u+g(u)\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), where \(\alpha\), \(\lambda\in\mathbb{R}\), \(g\in C(\mathbb{R})\) is a superlinear and subcritical function, and \(\Omega\) is a domain in \(\mathbb{R}^2\). We study the existence of positive solutions with respect to the values of the parameters \(\alpha\) and \(\lambda\), and according that \(0\in\Omega\) or \(0\in \partial \Omega\), and that \(\Omega\) is an exterior domain or not.