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July/August 2013 Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations
Yihong Du, Zongming Guo
Adv. Differential Equations 18(7/8): 737-768 (July/August 2013).

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).

Citation

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Yihong Du. Zongming Guo. "Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations." Adv. Differential Equations 18 (7/8) 737 - 768, July/August 2013.

Information

Published: July/August 2013
First available in Project Euclid: 20 May 2013

zbMATH: 1275.35102
MathSciNet: MR3086673

Subjects:
Primary: 35B45 , 35J40

Rights: Copyright © 2013 Khayyam Publishing, Inc.

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Vol.18 • No. 7/8 • July/August 2013
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