Finite Morse-Index Solutions and Asymptotics of Weighted Nonlinear Elliptic Equations

By introducing a suitable setting, we study the behavior of finite Morse-index solutions of the equation... We show that under our chosen setting for the finite Morse-index theory of (1), the stability of a solution to (1) is unchanged under various natural transformations. This enables us to reveal two critical values of the exponent p in (1) that divide the behavior of finite Morse-index solutions of (1), which in turn yields two critical powers for (2) through the transformation. The latter appear difficult to obtain by working directly with (2).