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A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials

About this Title

Florica C. Cîrstea, School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 227, Number 1068
ISBNs: 978-0-8218-9022-6 (print); 978-1-4704-1429-0 (online)
DOI: https://doi.org/10.1090/memo/1068
Published electronically: June 24, 2013
Keywords: Nonlinear elliptic equations, isolated singularities, regular variation theory, inverse square potentials, removable singularities
MSC: Primary 35J60, 35B40; Secondary 35J25, 35B33

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Table of Contents

Chapters

• 1. Introduction
• 2. Main results
• 3. Radial solutions in the power case
• 4. Basic ingredients
• 5. The analysis for the subcritical parameter
• 6. The analysis for the critical parameter
• 7. Illustration of our results
• A. Regular variation theory and related results

Abstract

In this paper, we consider semilinear elliptic equations of the form \begin{equation}\tag {0.1} -\Delta u- \frac {\lambda }{|x|^2}u +b(x)\,h(u)=0\quad \mbox {in } \Omega \setminus \{0\}, \end{equation} where $\lambda$ is a parameter with $-\infty <\lambda \leq (N-2)^2/4$ and $\Omega$ is an open subset in $\mathbb {R}^N$ with $N\geq 3$ such that $0\in \Omega$. Here, $b(x)$ is a positive continuous function on $\bar \Omega \setminus \{0\}$ which behaves near the origin as a regularly varying function at zero with index $\theta$ greater than $-2$. The nonlinearity $h$ is assumed continuous on $\mathbb {R}$ and positive on $(0,\infty )$ with $h(0)=0$ such that $h(t)/t$ is bounded for small $t>0$. We completely classify the behaviour near zero of all positive solutions of (0.1) when $h$ is regularly varying at $\infty$ with index $q$ greater than $1$ (that is, $\lim _{t\to \infty } h(\xi t)/h(t)=\xi ^q$ for every $\xi >0$). In particular, our results apply to (0.1) with $h(t)=t^q (\log t)^{\alpha _1}$ as $t\to \infty$ and $b(x)=|x|^\theta (-\log |x|)^{\alpha _2}$ as $|x|\to 0$, where $\alpha _1$ and $\alpha _2$ are any real numbers.

We reveal that the solutions of (0.1) generate a very complicated dynamics near the origin, depending on the interplay between $q$, $N$, $\theta$ and $\lambda$, on the one hand, and the position of $\lambda$ with respect to $0$ and $(N-2)^2/4$, on the other hand. Our main results for $\lambda =(N-2)^2/4$ appear here for the first time, as well as for the case $\lambda <0$. We establish a trichotomy of positive solutions of (0.1) under optimal conditions, hence generalizing and improving through a different approach a previous result with Chaudhuri on (0.1) with $0<\lambda <(N-2)^2/4$ and $b=1$. Moreover, recent results of the author with Du on (0.1) with $\lambda =0$ are here sharpened and extended to any $-\infty <\lambda <(N-2)^2/4$. In addition, we unveil a new single-type behaviour of the positive solutions of (0.1) specific to $0<\lambda <(N-2)^2/4$. We also provide necessary and sufficient conditions for the existence of positive solutions of (0.1) that are comparable with the fundamental solutions of \[ -Δu- \frac{𝜆}|x|^{2}u=0 in

\mathbb{R}^{N}∖{0}.\] In particular, for $b=1$ and $\lambda =0$, we find a sharp condition on $h$ such that the origin is a removable singularity for all non-negative solutions of (0.1), thus addressing an open question of Vázquez and Véron.