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

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.