Work

A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core

Year: 2025

Type: article

Abstract: We consider the equation $$\begin{aligned} -\Delta u+u=Q_\varepsilon (x)|u|^{p-2}u,\quad u\in H^1(\mathbb {R}^N), \end{aligned}$$ - Δ u + u = Q ε ( x ) | u | p - 2 u , u ∈ H 1 ( R N ) , where $$Q_\varepsilon $$ Q ε takes the value 1 on each ball $$B_\varepsilon (y)$$ B ε ( y ) , $$y\in \mathbb {Z}^N$$ y ∈ Z N , and the value $$-1$$ - 1 elsewhere. We establish the existence of a least energy solution for each $$\varepsilon \in (0,\frac{1}{2})$$ ε ∈ ( 0 , 1 2 ) and show that their $$H^1$$ H 1 and $$L^p$$ L p norms concentrate locally at points of $$\mathbb {Z}^N$$ Z N as $$\varepsilon \rightarrow 0$$ ε → 0 .

Source: Partial Differential Equations and Applications

DOI: 10.1007/s42985-025-00352-z

Authors: Mónica Clapp, Alberto Saldaña, Andrzej Szulkin

Institutions: Universidad Nacional Autónoma de México, Stockholm University

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: Royal Swedish Academy of Sciences, Consejo Nacional de Ciencia y Tecnología, Stockholms Universitet, Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México