On a singular elliptic equation
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- by Wei Ming Ni PDF
- Proc. Amer. Math. Soc. 88 (1983), 614-616 Request permission
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 ${C_1}|x{|^\sigma } \geqslant K(x) \geqslant {C_2}|x{|^\sigma }$ 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
- Patricio Aviles, On isolated singularities in some nonlinear partial differential equations, Indiana Univ. Math. J. 32 (1983), no. 5, 773–791. MR 711867, DOI 10.1512/iumj.1983.32.32051
- B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598. MR 615628, DOI 10.1002/cpa.3160340406
- P.-L. Lions, Isolated singularities in semilinear problems, J. Differential Equations 38 (1980), no. 3, 441–450. MR 605060, DOI 10.1016/0022-0396(80)90018-2
- Wei Ming 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), no. 4, 493–529. MR 662915, DOI 10.1512/iumj.1982.31.31040
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 88 (1983), 614-616
- MSC: Primary 35J60
- DOI: https://doi.org/10.1090/S0002-9939-1983-0702285-0
- MathSciNet review: 702285