Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations

TL;DR

This paper investigates the behavior of finite Morse index solutions to a class of weighted nonlinear elliptic equations, revealing critical exponents that determine solution stability and asymptotic behavior, using transformations to relate to Schrödinger equations with Hardy potential.

Contribution

It introduces a new setting for finite Morse index theory of weighted elliptic equations, identifying critical exponents that influence solution stability and asymptotics, and connects these to Schrödinger equations with Hardy potential.

Findings

Identified two critical exponents for solution behavior.

Established stability invariance under natural transformations.

Revealed asymptotic properties of solutions based on these exponents.

Abstract

By introducing a suitable setting, we study the behavior of finite Morse index solutions of the equation \[ -\{div} (|x|^\theta \nabla v)=|x|^l |v|^{p-1}v \;\;\; \{in $\Omega \subset \R^N \; (N \geq 2)$}, \leqno(1) \] where $p>1$, $\theta, l\in\R^1$ 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 Schr\"odinger equation with Hardy potential $$-\Delta u=|x|^\alpha |u|^{p-1}u+\frac{\ell}{|x|^2} u \;\; \{in $\Omega \subset \R^N \;\; (N \geq 2)$}, \leqno{(2)}$$ 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).