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 Ω⊆Rn, −Δu=μ/|x|αu2∗α-1+f(x)g(u) in Ω, u>0 in Ω, u=0 on ∂Ω where μ≥0, 0≤α≤2, 2α∗=2(n-α)/(n-2), f:Ω→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 Ip for which the embedding W1,p̅0(Ω)↪Lp(Ω,f) is compact. When p=2, α=2, f∈I2 and 0≤μ>(½(n−2))2, we prove that the linear problem has H10-discrete spectrum. By improving the Hardy inequality we show that for f belonging to a certain subclass of I2, the first eigenvalue goes to a positive number as μ approaches (½(n−2))2. Furthermore, when g is superlinear, we show that for the same subclass of I2, the functional corresponding to the differential equation satisfies the Palais-Smale condition if α=2 and a Brezis-Nirenberg type of phenomenon occurs for the case 0≤α<2