The Inverse Power Method for Semilinear Elliptic Equations

Abstract

In this note we will discuss an iterative procedure that is very useful for both the numerical and theoretical study of nonlinear eigenvalue problems

$$\begin{gathered}Lu \equiv {\partial _i}\left( {{a_{ij}}{\partial _j}u} \right) - a\left( x \right)u = - \lambda f\left( {x,u} \right)\quad in\,D \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad u = 0\;\;\;\;\;\;\;\;\;\;on\;\partial D \hfill \\\end{gathered}$$

((1))

where D is a bounded domain in R ⁿ and L is self-adjoint and uniformly elliptic with ${a_{ij}} \in {C^0}\left( {\bar D} \right)\;and\,a(x) \geqslant 0,\;a \in {L^\infty }$.