Schrodinger Equation With Multiparticle Potential and Critical Nonlinearity

We study the existence and non-existence of ground states for the Schrodinger equations -Delta u - lambda Sigma(i<j) u/vertical bar x(i) - x(j)vertical bar(2) = vertical bar u vertical bar(2)* -(2)u, x = (x(1), ... , x(m)) is an element of R-mN, and -Delta u -lambda u/vertical bar y vertical bar(2) = vertical bar u vertical bar(2*) (-2)u, x = (y, z) is an element of R-N. In both cases we assume lambda not equal 0 and lambda < (lambda) over bar, where (lambda) over bar is the Hardy constant corresponding to the problem