Advances in Differential Equations

Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations

Yihong Du and Zongming Guo

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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).

Article information

Source
Adv. Differential Equations Volume 18, Number 7/8 (2013), 737-768.

Dates
First available in Project Euclid: 20 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1369057712

Mathematical Reviews number (MathSciNet)
MR3086673

Zentralblatt MATH identifier
1275.35102

Subjects
Primary: 35B45: A priori estimates 35J40: Boundary value problems for higher-order elliptic equations

Citation

Du, Yihong; Guo, Zongming. Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations. Adv. Differential Equations 18 (2013), no. 7/8, 737--768. https://projecteuclid.org/euclid.ade/1369057712.


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