Chaudhuri, Nirmalendu ; Ramaswamy, Mythily
(2001)
*Existence of positive solutions of some semilinear elliptic equations with singular coefficients*
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 131
(6).
pp. 1275-1295.
ISSN 0308-2105

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Official URL: http://journals.cambridge.org/action/displayAbstra...

Related URL: http://dx.doi.org/10.1017/S0308210500001396

## Abstract

In this paper we consider the semilinear elliptic problem in a bounded domain Ω⊆R^{n}, −Δu=μ/|x|^{α}u^{2∗α-1}+f(x)g(u) in Ω, u>0 in Ω, u=0 on ∂Ω where μ≥0, 0≤α≤2, 2_{α∗}=2(n-α)/(n-2), f:Ω→R^{+} is measurable, f>0 a.e, having a lower-order singularity than |x|^{−2} at the origin, and g:R→R is either linear or superlinear. For 1>p>n, we characterize a class of singular functions I_{p} for which the embedding W^{1,p̅}_{0}(Ω)↪L^{p}(Ω,f) is compact. When p=2, α=2, f∈I_{2} and 0≤μ>(½(n−2))^{2}, we prove that the linear problem has H^{1}_{0}-discrete spectrum. By improving the Hardy inequality we show that for f belonging to a certain subclass of I_{2}, the first eigenvalue goes to a positive number as μ approaches (½(n−2))^{2}. Furthermore, when g is superlinear, we show that for the same subclass of I_{2}, the functional corresponding to the differential equation satisfies the Palais-Smale condition if α=2 and a Brezis-Nirenberg type of phenomenon occurs for the case 0≤α<2.

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Source: | Copyright of this article belongs to Cambridge University Press. |

ID Code: | 62283 |

Deposited On: | 20 Sep 2011 09:31 |

Last Modified: | 20 Sep 2011 09:31 |

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