Year: 2026
Type: article
Abstract: Let $$s\in (0,1)$$ s ∈ ( 0 , 1 ) , $$\varepsilon >0$$ ε > 0 and let $$\Omega $$ Ω be a bounded smooth domain. Given the problem $$\begin{aligned} \varepsilon ^{2s}(-\Delta )^{s} u + V(x)u = |u|^{p-1}u \quad \text{ in } \; \Omega , \end{aligned}$$ ε 2 s ( - Δ ) s u + V ( x ) u = | u | p - 1 u in Ω , with Dirichlet boundary conditions and $$1 1 < p < ( n + 2 s ) / ( n - 2 s ) , we analyze the existence of positive multi-peak solutions concentrating, as $$\varepsilon \rightarrow 0$$ ε → 0 , to one or several points of $$\Omega $$ Ω . Under suitable conditions on V , we construct positive solutions concentrating at any prescribed set of its non degenerate critical points. Furthermore, we prove existence and non existence of clustering phenomena around local maxima and minima of V , respectively. The proofs rely on a Lyapunov-Schmidt reduction where three effects need to be controlled: the potential, the boundary and the interaction among peaks. The slow decay of the associated ground-state demands very precise asymptotic expansions.
Authors: Maria Medina, Jing Wu
Institutions: Universidad Autónoma de Madrid, University of Macau
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4416888590
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