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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a singular elliptic equation


Author: Wei Ming Ni
Journal: Proc. Amer. Math. Soc. 88 (1983), 614-616
MSC: Primary 35J60
MathSciNet review: 702285
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Abstract: In this paper, we study the singular elliptic equation $ Lu + K(x){u^p} = 0$, where $ L$ is a uniformly elliptic operator of divergence form, $ p > 1$ and $ K(x)$ has a singularity at the origin. We prove that this equation does not possess any positive (local) solution in any punctured neighborhood of the origin if there exist two constants $ {C_1}$, $ {C_2}$ such that $ {C_1}\vert x{\vert^\sigma } \geqslant K(x) \geqslant {C_2}\vert x{\vert^\sigma }$ near the origin for some $ \sigma \leqslant - 2$ (with no other condition on the gradient of $ K$ ). In fact, an integral condition is derived.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0702285-0
PII: S 0002-9939(1983)0702285-0
Keywords: Singular elliptic equation, nonexistence
Article copyright: © Copyright 1983 American Mathematical Society