GEOMETRIC RESULT FOR THE ELLIPTIC PROBLEM WITH NONLINEARITY CROSSING THREE EIGENVALUES

Abstract

We investigate the number of the solutions for the ellip-
tic boundary value problem. We obtain a theorem which shows the
existence of six weak solutions for the elliptic problem with jumping
nonlinearity crossing three eigenvalues. We get this result by using
the geometric mapping defined on the finite dimensional subspace.
We use the contraction mapping principle to reduce the problem
on the infinite dimensional space to that on the finite dimensional
subspace. We construct a three dimensional subspace with three
axis spanned by three eigenvalues and a mapping from the finite
dimensional subspace to the one dimensional subspace.