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 Omega sube Rn,

-Deltau = (mu/|x|alpha)u2*alpha - 1 + f(x)g(u) in Omega,

u > 0 in Omega,

u = 0 on partOmega

where mu ge 0, 0 le alpha le 2, 2alpha* := 2(n - alpha)/(n - 2), f : Omega 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 Tp for which the embedding W01,p(Omega) rarr Lp(Omega, f) is compact. When p = 2, alpha = 2, f isin T2 and 0 le mu < (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 mu 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 alpha = 2 and a Brezis-Nirenberg type of phenomenon occurs for the case 0 le alpha < 2.

Document Type: Research article

Publication date: 2001-12-14

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