Second order elliptic equations with degenerate weight

Abstract

We consider the eigenvalue problem: − Δ u − q u = λ ω u , u ∈ H ˙ 1 , 2 ( Ω ) - \Delta u - qu = \lambda \omega u,u \in \dot {H}^{1,2}(\Omega ) , in a smooth bounded domain Ω ⊂ R n \Omega \subset {{\mathbf {R}}^n} . We allow − Δ − q - \Delta - q to have negative spectrum and assume ω ≥ 0 \omega \geq 0 in Ω , ω ≡ 0 \Omega ,\omega \equiv 0 in a subdomain of Ω \Omega . Under suitable regularity conditions, we establish several results for the spectrum of this problem. In particular, we give: a min.max. formula for λ \lambda ; a precise estimate on the number of negative λ \lambda ; an estimate for the location of negative λ \lambda . An example concludes the paper.