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# memo_has_moved_text();A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials

Florica C. Cîrstea

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: http://dx.doi.org/10.1090/memo/1068
Published electronically: June 24, 2013
Keywords:Nonlinear elliptic equations, isolated singularities, regular variation theory, inverse square potentials

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Chapters

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

### Abstract

In this paper, we consider semilinear elliptic equations of the form

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 $\Omega \setminus \{0\}$ 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 [[eqref]]one 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 [[eqref]]one 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 [[eqref]]one 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 [[eqref]]one under optimal conditions, hence generalizing and improving through a different approach a previous result with Chaudhuri on [[eqref]]one 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 [[eqref]]one 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

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 [[eqref]]one, thus addressing an open question of Vázquez and Véron.