The authors consider the solutions of the Laplace and the Poisson equation on bounded open subsets \(\Omega\) of \(\mathbb{R}^n\) with \(n\geq 2\). Usually, solutions are obtained with Newtonian potential kernel and Newtonian potential operators. However, new solutions can be constructed with Hammerstein integral operators that involve kernels and Green’s functions. The authors also provide results for an eigenvalue problem for the Laplace equation using their new approach that are not obtainable using the weight Newtonian potential operator together with the Krein-Rutman theorem when \(n=2\). Remarkably, neither connectedness on \(\Omega\) nor smoothness of the boundary \(\partial \Omega\) is required.