Existence of positive solutions of some semilinear elliptic equations with singular coefficients

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
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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