Asymptotic profiles for a nonlinear Schr\"odinger equation with critical combined powers nonlinearity

TL;DR

This paper analyzes the asymptotic behavior of positive ground state solutions to a nonlinear Schrödinger equation with combined critical and subcritical powers, revealing dimension-dependent rescaling limits as the parameter tends to zero.

Contribution

It provides a precise asymptotic characterization of ground state solutions' rescaling behavior for different space dimensions, extending understanding of critical nonlinear Schrödinger equations.

Findings

Rescaling limits depend on the space dimension N.

Solutions converge to a critical Emden-Fowler solution as λ→0.

The asymptotic profile is sharply characterized for N=3, 4, and ≥5.

Abstract

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schr\"odinger equation $$ -\Delta u+ u=u^{2^*-1}+\lambda u^{q-1} \quad {\rm in} \ \ \mathbb{R}^N, $$ where $N\ge 3$ is an integer, $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, $2<q<2^*$ and $\lambda>0$ is a parameter. It is known that as $\lambda\to 0$, after a rescaling the ground state solutions of the equation converge to a particular solution of the critical Emden-Fowler equation $-\Delta u=u^{2^*-1}$. We establish a sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension $N=3$, $N=4$ or $N\ge 5$.