Solutions of Hammerstein integral equations via a variational principle
The authors study solutions of the Hammerstein equation \[ u(x)= \int_\Omega k(x,y)f \bigl(y,u(y)\bigr)\,dy \] as critical points of the corresponding energy functional on a suitable Hilbert space. The main tool is a new variational principle of \textit{B. Ricceri} [J. Comput. Appl. Math. 113, No. 1--2, 401--410 (2000; Zbl 0946.49001)] which also applies to kernel functions \(k\) of changing sign. The abstract result is illustrated by means of an application to the polyharmonic equation \((-\Delta)^mu(x)= f(x,u(x))\).
scientific article; zbMATH DE number 2114499