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Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity

Vitaly Moroz Orcid Logo, Shiwang Ma

Mathematische Zeitschrift, Volume: 304, Issue: 13

Swansea University Author: Vitaly Moroz Orcid Logo

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DOI (Published version): 10.1007/s00209-023-03271-0

Abstract

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation−Δu+u=u2∗−1+λuq−1inRN,(Pλ)where N≥3 is an integer, 2∗=2NN−2 is the Sobolev critical exponent, 2<q<2∗ and λ>0 is a parameter. It is known that as λ→0, after a rescaling the ground state solu...

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Published in: Mathematische Zeitschrift
ISSN: 0025-5874 1432-1823
Published: Springer Nature 2023
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URI: https://cronfa.swan.ac.uk/Record/cronfa63145
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Abstract: We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation−Δu+u=u2∗−1+λuq−1inRN,(Pλ)where N≥3 is an integer, 2∗=2NN−2 is the Sobolev critical exponent, 2<q<2∗ and λ>0 is a parameter. It is known that as λ→0, after a rescaling the ground state solutions of (Pλ) converge to a particular solution of the critical Emden-Fowler equation −Δu=u2∗−1. We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension N=3, N=4 or N≥5. We also discuss a connection of these results with a mass constrained problem associated to (Pλ). Unlike previous work of this type, our method is based on the Nehari-Pohožaev manifold minimization, which allows to control the L2 norm of the groundstates.
Keywords: Nonlinear Schrödinger equation, critical Sobolev exponent, concentration compactness, asymptotic behaviour
College: Faculty of Science and Engineering
Funders: S.M. was supported by National Natural Science Foundation of China (Grant Nos.11571187, 11771182).
Issue: 13

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