The authors investigate existence results for the nonlinear elliptic problem \[ \Delta u+\varphi(x,u)=0, \quad x\in\Omega,\qquad fu|_{\partial\Omega}=0. \] Here, \(\Omega\subset\mathbb {R}^d\), \(d\geq 3\), is a (possibly unbounded) domain and \(\varphi:\Omega\times (0,\infty)\to (0,\infty)\) is assumed to be continuous and nonincreasing with respect to the second variable. They prove the existence and uniqueness of a positive solution \(u\in C(\overline{\Omega})\), and in the case when \(\varphi\in C^{\alpha}_{\text{loc}}(\Omega\times (0,\infty))\), they show the solution \(u\in C^{2+\alpha}_{\text{loc}}(\Omega)\cap C(\overline{\Omega})\). Finally, they establish some estimates on the solution, including the lower bound \(u(x)\geq c\theta(x)\), for all \(x\in\Omega\), where \(\theta(x)=\min\{1,G(x,x_0)\}\). Here, \(x_0\in\Omega\) is fixed, \(c\) is a positive constant, and \(G\) is the Green’s function.