Blowing-up solutions for a nonlocal Liouville type equation

2022/04/12 by Matteo Cozzi, Cozzi, Matteo, A. Fernández +1
Mathematics · #35B25 #35B44 #35R11 #35S15 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2204.05837

openalex publication_date 2022/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the nonlocal Liouville type equation (-Δ)^(1)/(2) u = ε κ(x) e^u, u gt; 0, in I, u = 0, in ℝ ∖ I, where I is a union of d ≥ 2 disjoint bounded intervals, κ is a smooth bounded function with positive infimum and ε > 0 is a small parameter. For any integer 1 ≤ m ≤ d, we construct a family of solutions (u_ε)_ε which blow up at m interior distinct points of I and for which ε ∫_I κe^u_ε → 2 m π, as ε → 0. Moreover, we show that, when d = 2 and m is suitably large, no such construction is possible.

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