Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [A] P. Aviles, On isolated singularities in some nonlinear partial differential equations, Indiana Univ. Math. J. (to appear). MR 711867 (84h:35053)
  • [G, S] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 36 (1981), 525-598. MR 615628 (83f:35045)
  • [L] P.-L. Lions, Isolated singularities in semilinear problems, J. Differential Equations 38 (1980), 441-450. MR 605060 (82g:35040)
  • [N] W.-M. Ni, On the elliptic equation $ \Delta u + K(x){u^{(n + 2)/(n - 2)}} = 0$, its generalizations and applications in geometry, Indiana Univ. Math. J. 31 (1982), 493-529. MR 662915 (84e:35049)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35J60

Retrieve articles in all journals with MSC: 35J60

Additional Information

Keywords: Singular elliptic equation, nonexistence
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society