Stable and singular solutions of the equation $Δu = 1/u$

Details Stable and singular solutions of the equation $Δu = 1/u$ Stable and singular solutions of the equation $Δu = 1/u$

2004-04-22

arxiv.org/abs/math/0404422v1

Mathematics

Analysis of PDEs

20 pages, no figures

Scientific paper

We study properties of the semilinear elliptic equation $\Delta u = 1/u$ on domains in $R^n$, 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.

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