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

TL;DR

This paper analyzes the asymptotic behavior of positive solutions to a scalar field equation with a small parameter, revealing how solutions transition between different regimes depending on the relation of p to the critical Sobolev exponent.

Contribution

It provides a complete characterization of all possible asymptotic regimes for solutions as the parameter approaches zero, based on the relation of p to the critical Sobolev exponent.

Findings

Solutions match different limiting equations depending on p relative to p*

For p<p*, solutions resemble the equation without the last term

For p>p*, solutions resemble the equation with epsilon=0

Abstract

We study the leading order behaviour of positive solutions of the equation -\Delta u +\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 $\varepsilon>0$ 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 $\varepsilon=0$. 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 $\varepsilon$ in a nontrivial way.