Mathematische Annalen

Mathematische

Math Ann 269, 119-135 (1984) АППвкП

© Springer-Verlag 1984

Singularities of Elliptic Equations with an Exponential Nonlinearity

Juan Luis Vazquez^ and Laurent Veron^

1 Matematicas, Universidad Autonoma de Madrid, Madrid 34, Spain

2 Département de Mathématiques, Université de Tours, Parc de Grandmont, F-37200 Tours, France

Introduction

This paper is concerned with the existence and the description of the isolated singularities of the solutions of the equation

- Au + g{u)=f, (0.1)

mQ' = Q — {0} where Q is an open subset of R^ containing 0, g a continuous non- decreasing real valued function defined on IR and / a continuous function.

When g(u) = u\u\'^~^, ^> U and/=0 many results concerning the singularities of the solutions of (0.1) have been given by Veron [11]. In particular he gave a full description of the isolated singularities when q^3:

i ) either u(x)/Log{l/\x\) converges as x-^0 to some constant с which can take any value, and и satisfies

- Au + u\u\^~'^=2ncôo in Щи), (0.2)

ii ) or |xp^^" ^^w(x) converges to a constant which can take only the two values

± ( 2 / ( q - l ) ) 2«'' - '' .

In this paper we are interested in the solutions и of (0.1) satisfying the limiting

growth condition ,-_ / ч л /a a\

limxw ( x ) = 0, (0.3)

x - >0

and we first prove the following isotropy result:/or any solution o/(0.1) satisfying

( 0 . 3 ) , the following limit

limu ( x ) / Log ( l / |x| ) , (0.4)

exists in lRu{ — oo, + oo}. Moreover if the limit (0.4) is zero then и is nonsingular in Q

In order to give a full description of и near 0 we suppose that g satisfies the loilowing technical condition

for any a >0, lim e~'''"^(r) and lim e'^''g{r) 1 exist in lRu{ + oo, — oo}, J