Existence and non-existence of solution for semilinear elliptic equation with harmonic potential and Sobolev critical/supercritical nonlinearities

In this paper, we study the existence and non-existence of the positive solution to a semilinear elliptic equation with the harmonic potential and power type nonlinearity. First, we obtain non-existence result in three space dimension with the Sobolev critical nonlinearity. Next, we prove that there exists a global bifurcation branch with the Sobolev supercritical nonlinearity.