TY - JOUR T1 - An Eigenvalue Problem on Negatively Curved Manifolds AU - Fang Hua Lin JO - Journal of Partial Differential Equations VL - 4 SP - 1 EP - 8 PY - 1991 DA - 1991/04 SN - 4 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5779.html KW - Eigenvalue problem KW - complete manifolds KW - negative curvature AB - Consider the eigenvalue problem: Δgu - λKu = 0 \quad in D where D is the unit disc of the complex plane, g is a complete metric conformal to the Poincaré metric on D, and K is the Gaussian curvature. It is shown that if λ > \frac{1}{2} (λ > \frac{1}{4}in the case of K ≤ 0), then the above problem has no positive solutions.