Positive solutions to multi-critical elliptic problems

In this paper, we investigate the existence of multiple positive solutions to the following multi-critical elliptic problem {-Delta u = lambda vertical bar u vertical bar(p-2)u + Sigma(k)(i=1)(vertical bar x vertical bar(-(N-alpha i)) * vertical bar u vertical bar 2i*)vertical bar u vertical bar(2i)*(-2)u in Omega, u is an element of H-0(1)(Omega) in connection with the topology of the bounded domain Omega subset of R-N, N >= 4, where lambda > 0, 2(i)* = N+alpha(i)/N-2 with N - 4 < alpha(i) < N, i = 1, 2, . . . , k are critical Hardy-Littlewood-Sobolev exponents and 2 < p < 2* = 2N/N-2. We show that there is lambda* > 0 such that if 0 < lambda < lambda* problem (0.1) possesses at least cat(Omega) (Omega) positive solutions. We also study the existence and uniqueness of positive solutions for the limit problem of (0.1).