Boundedness and monotonicity of principal eigenvalues for boundary value problems with indefinite weight functions

Abstract

We study the principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem: −Δu(x)=λg(x)u(x), x∈D;(∂u/∂n)(x)+αu(x)=0, x∈∂D, where Δ is the standard Laplace operator, D is a bounded domain with smooth boundary, g:D→ℝ is a smooth function which changes sign on D and α∈ℝ. We discuss the relation between α and the principal eigenvalues.