SIX SOLUTIONS FOR THE SEMILINEAR WAVE EQUATION WITH NONLINEARITY CROSSING THREE EIGENVALUES

Abstract

We get a theorem which shows the existence of at least six solutions for the semilinear wave equation with nonlinearity crossing three eigenvalues. We obtain this result by the variational reduction method and the geometric mapping defined on the finite dimensional subspace. We use a contraction mapping principle to reduce the problem on the infinite dimensional space to that on the finite dimensional subspace. We construct a three-dimensional sub-space with three axes spanned by three eigenvalues and a mapping from the finite dimensional subspace to the one-dimensional sub-space.