2025/02/20 by Serena Dipierro, Dipierro, Serena, Xifeng Su +5
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2502.14483
openalex publication_date 2025/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the mixed local/nonlocal semilinear equation -ε2Δu +ε2s(-Δ)s u +u=up in Ω with zero Dirichlet datum, where ε>0 is a small parameter, s∈(0,1), p∈(1,(n+2)/(n-2)) and Ω is a smooth, bounded domain. We construct a family of solutions that concentrate, as ε→ 0, at an interior point of Ω having uniform distance to ∂Ω (this point can also be characterized as a local minimum of a nonlocal functional). In spite of the presence of the Laplace operator, the leading order of the relevant reduced energy functional in the Lyapunov-Schmidt procedure is polynomial rather than exponential in the distance to the boundary, in light of the nonlocal effect at infinity. A delicate analysis is required to establish some uniform estimates with respect to ε, due to the difficulty caused by the different scales coming from the mixed operator.