Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy-Sobolev Critical Exponents

The following semi-linear elliptic equations involving Hardy-Sobolev critical exponents -Delta u - mu u/vertical bar x vertical bar(2) = vertical bar u vertical bar(2)*((s)-2)/vertical bar x vertical bar s u + g(x, u), x is an element of Omega \ {0}, u = 0, x partial derivative Omega O have been investigated, where O is an open-bounded domain in R-N(N >= 3), with a smooth boundary partial derivative Omega, 0 is an element of Omega, 0 = mu < <(mu)over bar>:= (N-2/2)(2), 0 <= s < 2, and 2*(s) = 2(N - s)/(N - 2) is the Hardy-Sobolev critical exponent. This problem comes from the study of standing waves in the anisotropic Schrodinger equation; it is very important in the fields of hydrodynamics, glaciology, quantum field theory, and statistical mechanics. Under some deterministic conditions on g, by a detailed estimation of the extremum function and using mountain pass lemma with (PS)(c) conditions, we obtained that: (a) If mu = <(mu)over bar>- 1, and lambda < lambda(1)(mu), then the above problem has at least a positive solution in H-0(1) (Omega); (b) If (mu) over bar - 1 < mu < (mu) over bar, then when lambda(*)(mu) < lambda < lambda(1)(mu), the above problem has at least a positive solution in H-0(1)(Omega); (c) if (mu) over bar - 1 < mu < (mu) over bar and Omega = B(0, R), then the above problem has no positive solution for lambda <= lambda(*)(mu). These results are extensions of E. Jannelli's research (g(x, u) = lambda u).