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

2021/08/03 by Shiwang Ma, Vitaly Moroz, Ma, Shiwang +1
Mathematics · #35B25 #35B40 #35J60 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2108.01421

openalex publication_date 2021/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation -Δu+ u=u^{2^*-1}+λu^q-1 \rm in ℝ^N, where N≥ 3 is an integer, 2^*=(2N)/(N-2) is the Sobolev critical exponent, 20 is a parameter. It is known that as λ→ 0, after a rescaling the ground state solutions of the equation converge to a particular solution of the critical Emden-Fowler equation -Δu=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≥ 5.

Citations

• Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case
• The double-power nonlinear Schr "odinger equation and its\n generalizations: uniqueness, non-degeneracy and applications

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