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

Authors: Chaudhuri N.; Ramaswamy M.

Source: Proceedings Section A: Mathematics - Royal Society of Edinburgh, Volume 131, Number 6, 14 December 2001 , pp. 1275-1295(21)

Publisher: Royal Society of Edinburgh

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Abstract:

In this paper we consider the semilinear elliptic problem in a bounded domain sube

Rⁿ,

u = (/|x|)u^{2* - 1} + f(x)g(u) in ,

u > 0 in ,

u = 0 on

where

0, 0

2, 2* := 2(n - )/(n - 2), f : rarr

R⁺ is measurable, f > 0 a.e, having a lower-order singularity than |x|^-2 at the origin, and g : R rarr

R is either linear or superlinear. For 1 < p < n, we characterize a class of singular functions T_p for which the embedding W₀^1,p() rarr

L^p(, f) is compact. When p = 2, = 2, f isin

T₂ and 0

< (½(n - 2))², we prove that the linear problem has H₀¹-discrete spectrum. By improving the Hardy inequality we show that for f belonging to a certain subclass of T₂, the first eigenvalue goes to a positive number as approaches (½(n - 2))². Furthermore, when g is superlinear, we show that for the same subclass of T₂, the functional corresponding to the differential equation satisfies the Palais-Smale condition if = 2 and a Brezis-Nirenberg type of phenomenon occurs for the case 0

< 2.

Document Type: Research article

Publication date: 2001-12-14