Asymptotic properties of ground states of scalar field equations with a vanishing parameter

Abstract

We study the leading order behaviour of positive solutions of the equation -\Delta u +\epsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in\mathbb R^N, where N\ge 3 , q>p>2 and when \epsilon >0 is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of p , q and N . The behavior of solutions depends sensitively on whether p is less, equal or bigger than the critical Sobolev exponent 2^\ast=\frac{2N}{N-2} . For p<2^\ast the solution asymptotically coincides with the solution of the equation in which the last term is absent. For p>2^\ast the solution asymptotically coincides with the solution of the equation with \epsilon =0 . In the most delicate case p=2^\ast the asymptotic behaviour of the solutions is given by a particular solution of the critical Emden–Fowler equation, whose choice depends on \epsilon in a nontrivial way.