Stable and singular solutions of the equation \(\Delta u = 1/u\). (English) Zbl 1129.35395
Summary: We study properties of the semilinear elliptic equation \(\Delta u = 1/u\) on domains in \(\mathbb{R}^n\), with an eye toward nonnegative singular solutions as limits of positive smooth solutions. We prove the nonexistence of such solutions in low dimensions when we also require them to be stable for the corresponding variational problem. The problem of finding singular solutions is related to the general study of singularities of minimal hypersurfaces in Euclidean space.
MSC:
35J60 | Nonlinear elliptic equations |
35D10 | Regularity of generalized solutions of PDE (MSC2000) |
35J20 | Variational methods for second-order elliptic equations |
47H11 | Degree theory for nonlinear operators |
47N20 | Applications of operator theory to differential and integral equations |
58E12 | Variational problems concerning minimal surfaces (problems in two independent variables) |