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

In this paper we consider the semilinear elliptic problem in a bounded domain Omega subset of or equal to R-n, -Deltau = u/\x\(alpha) u 2*alpha(-1) + f(x)g(u) in Omega, u > 0 in Omega, u = 0 on partial derivativeOmega where mu greater than or equal to 0, 0 less than or equal to alpha less than or equal to 2, 2(alpha)*, := 2(n - alpha)/(n - 2), f : Omega --> 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 S-p for which the embedding W-0(1.p)(Omega) hooked right arrow L-p(Omega, f) is compact. When p = 2, alpha = 2, 0 f epsilon S-2 and 0 less than or equal to mu (1/2(n-2))(2), we prove that the linear problem has H-0(1)-discrete 2 spectrum. By improving the. Hardy inequality we show that for f belonging to a certain subclass Of S-2, the first eigenvalue goes to a positive number as mu approaches (1/2(n-2))(2). Furthermore. when g is superlinear, we show that for the same subclass 2 Of S-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 less than or equal to alpha less than or equal to 2.