Asymptotics for semilinear elliptic equations with supercritical nonlinearities in annular domains

Abstract

We study radially symmetric solutions of the equation Δu+u^p=0 in annular domains {a<|x|<1} in R³. In particular we are interested in the behaviour of solutions when the power p tends to the critical Sobolev exponent 5 in R³ and simultaneously, the inner radius shrinks to zero. When p $\searrow$ 5, two forms of behaviour are observed, depending on whether (p−5)²«a or a«(p−5)².