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

Mónica Clapp (), Alberto Saldaña () and Andrzej Szulkin ()
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Mónica Clapp: Universidad Nacional Autónoma de México, Instituto de Matemáticas
Alberto Saldaña: Universidad Nacional Autónoma de México, Circuito Exterior, Instituto de Matemáticas
Andrzej Szulkin: Stockholm University, Department of Mathematics

Partial Differential Equations and Applications, 2025, vol. 6, issue 6, 1-11

Abstract: 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 .

Keywords: Schrödinger equation; Periodic self-focusing core; Concentration of least energy solutions; 35J61; 35J20; 35B40; 35B44 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s42985-025-00352-z

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