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

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## Abstract

In this paper, we consider the semilinear elliptic problem in a bounded domain \Omega \subseteq $R^n$, $-{\Delta}u = \frac{\mu}{|x|^{\alpha}}u^{2^{\ast}_{\alpha}-1} + f(x)g(u) in \Omega$ u>0 in \Omega u=0 on \partial\Omega where $\mu \geq 0$, $0 \leq \alpha \leq 2$, $2^{\asterik}_{\alpha} : = 2(n - \alpha ) / (n - 2)$, $f : \Omega \rightarrow R^+$ is measurable, f > 0 a.e, having a lower-order singularity than $|x|^{-2}$ at the origin, and $g : R \rightarrow R$ is either linear or superlinear. For 1 < p < n, we characterize a class of singular functions $\Im_p$ for which the embedding $W_0^{1,p} (\Omega) \hookrightarrow L^p (\Omega, f)$ is compact. When p = 2, $\alpha = 2$, $f \in \Im_2$ and $0 \leq \mu < (\frac {1}{2}\(n-2))^2$, we prove that the linear problem has $H_0^1$ –discrete spectrum. By improving the Hardy inequality we show that for f belonging to a certain subclass of $\Im _2$, the first eigenvalue goes to a positive number as \mu approaches $(\frac{1}{2}\(n-2))^2$ Furthermore, when g is superlinear, we show that for the same subclass of $\Im_2$, the functional corresponding to the differential equation satisfies the Palais-Smale condition if $\alpha = 2$ and a Brezis-Nirenberg type of phenomenon occurs for the case $0 \leq \alpha < 2$.

Item Type: | Journal Article |
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Publication: | Proceedings of the Royal Society of Edinburgh: Mathematics |

Publisher: | Royal Society of Edinburgh |

Additional Information: | Copyright of this article belongs to Royal Society of Edinburgh. |

Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |

Date Deposited: | 01 Aug 2008 |

Last Modified: | 19 Sep 2010 04:47 |

URI: | http://eprints.iisc.ac.in/id/eprint/15116 |

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