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 -Δu +\varepsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in\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 $p^{\ast }=\frac{2N}{N-2}$. For $p<p^{\ast }$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p>p^{\ast }$ the solution asymptotically coincides with the solution of the equation with $\epsilon =0$. In the most delicate case $p=p^{\ast }$ 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.

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