Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 87, pp. 1-15.

Title: Solutions and eigenvalues of Laplace's equation on bounded open sets
       
Authors: Guangchong Yang (Chengdu Univ. of Information Tech., Sichuan, China)
         Kunquan Lan (Ryerson Univ., Toronto, Ontario, Canada)

Abstract:
We obtain solutions for  Laplace's  and Poisson's equations on 
bounded open subsets of
R<sup>n,</sup> (n&ge;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 equation
are different from the previous results that use the Newtonian potential operator and
require n&ge;3.
As a special case of the eigenvalue problems, we provide a result under
an easily verifiable condition on the weight function when n&ge;3.
This result cannot be obtained by using the Newtonian potential operator.

Submitted July 12, 2021. Published October 18, 2021.
Math Subject Classifications: 35J05, 31A05, 31B05, 35J08, 47A75.
Key Words: Eigenvalue; Laplace's equation; Poisson's equation; Green's function;
           Hammerstein integral operator.