Analysis of search-extension method for finding multiple solutions of nonlinear problem (Q2481274)

A search-extension method (SEM) is presented to the solution of the semilinear elliptic equation \(\Delta u+ f(x,u)= 0\) in \(\Omega\), \(u= 0\) on \(\Gamma\), where \(\Omega\) is a \(d\)-dimensional bounded domain with a regular boundary \(\Gamma\). SEM includes two basic parts: first search out better initial values of all solutions in a linear subspace spanned by several eigenbases, then further approximate by an effective extension iterative scheme in a larger subspace. The purpose of this paper is to establish a theoretical foundation of SEM.