Work

Asymptotic profiles for Choquard equations with combined attractive nonlinearities: the locally critical case

Year: 2025

Type: article

Abstract: We study asymptotic behavior of positive ground state solutions of the nonlinear Choquard equation with a Sobolev critical attractive local perturbation where $$N\ge 3$$ N ≥ 3 is an integer, $$p\in \big (\frac{N+\alpha }{N}, \frac{N+\alpha }{N-2}\big )$$ p ∈ ( N + α N , N + α N - 2 ) , $$2^*=\frac{2N}{N-2}$$ 2 ∗ = 2 N N - 2 is the Sobolev critical exponent, $$I_\alpha $$ I α is the Riesz potential of order $$\alpha \in (0,N)$$ α ∈ ( 0 , N ) and $$\varepsilon >0$$ ε > 0 is a parameter. We show that as $$\varepsilon \rightarrow \infty $$ ε → ∞ , after suitable rescalings the ground state solutions $$u_{\varepsilon }$$ u ε of $$(P_\varepsilon )$$ ( P ε ) converge to a particular solution of the critical local Emden–Fowler equation. The rescalings are implicit and depend in a non-trivial way on the exponent p and the space dimension $$N=3,4$$ N = 3 , 4 or $$N\ge 5$$ N ≥ 5 . We establish a sharp asymptotic characterisation of such rescalings, as well as the blow-up rates or asymptotics of the $$L^2$$ L 2 and other relevant norm of $$u_{\varepsilon }$$ u ε . As a follow up of our main results, we also obtain the existence, multiplicity and asymptotic behaviour of positive normalized solutions of a mass constrained problem associated to $$(P_\varepsilon )$$ ( P ε ) with mass normalization constraint $$\int _{\mathbb {R}^N}|u|^2=c^2$$ ∫ R N |

Source: Calculus of Variations and Partial Differential Equations

DOI: 10.1007/s00526-025-02986-8

Authors: Shiwang Ma, Vitaly Moroz

Institutions: Nankai University, Swansea University

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190

Funder: National Natural Science Foundation of China