Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations

Abstract

By introducing a suitable setting, we study the behavior of finite Morse-index solutions of the equation \begin{equation} -\operatorname{div} (|x|^\theta \nabla v)=|x|^l |v|^{p-1}v \;\;\; \mbox{in $\Omega \subset \mathbb{R}^N \; (N \geq 2)$}, \tag{1} \end{equation} where $p>1$, $\theta, l\in\mathbb{R}^1$ with $N+\theta>2$, $l-\theta>-2$, and $\Omega$ is a bounded or unbounded domain. Through a suitable transformation of the form $v(x)=|x|^\sigma u(x)$, equation (1) can be rewritten as a nonlinear Schrödinger equation with Hardy potential \begin{equation} -\Delta u=|x|^\alpha |u|^{p-1}u+\frac{\ell}{|x|^2} u \;\; \mbox{in $\Omega \subset \mathbb{R}^N \;\; (N \geq 2)$}, \tag{2} \end{equation} where $p>1$, $\alpha \in (-\infty, \infty)$, and $\ell \in (-\infty, (N-2)^2/4)$. 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).