Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy-Sobolev-Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy-Sobolev-Maz'ya term: -Delta u - lambda u/vertical bar y vertical bar(2) - vertical bar u vertical bar(pt-1)u/vertical bar y vertical bar(t) + mu f(x), x is an element of Omega where Omega is a bounded domain in R-N (N >= 2), 0 is an element of Omega, x = (y, z), is an element of R-k x RN-k and p(t) = N+2-2t/N-2 (0 <= t <= 2). For f(x) is an element of C-1 (<(Omega) over bar>)/{0}, we show that there exists a constant mu* > 0 such that the problem possesses at least two positive solutions if mu is an element of (0, mu*) and at least one positive solution if mu = mu*. Furthermore, there are no positive solutions if mu is an element of (mu*, +infinity).