Solutions and eigenvalues of Laplace's equation on bounded open sets

Abstract

We obtain solutions for Laplace's and Poisson's equations on bounded open subsets of \(R^n\) (\(n\geq 2)\),  via Hammerstein integral operators involving kernels and Green's functions, respectively. The new solutions are different from the previous ones obtained by the well-known Newtonian potential kernel and the Newtonian potential operator. Our results on eigenvalue problems of Laplace's equationare different from the previous results that use the Newtonian potential operator and require \(n\geq 3\). As a special case of the eigenvalue problems, we provide a result under an easily verifiable condition on the weight function when \(n\geq 3\). This result cannot be obtained by using the Newtonian potential operator. For more information see https://ejde.math.txstate.edu/Volumes/2021/87/abstr.html