Second order elliptic equations with degenerate weight

Allegretto, W.

doi:10.1090/S0002-9939-1989-0977929-4

Second order elliptic equations with degenerate weight
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Proc. Amer. Math. Soc. 107 (1989), 989-998 Request permission

Abstract:

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

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