Existence of nonnegative solutions for fractional Schrödinger equations with Neumann condition

2022/11/30 by Hamilton Bueno, Bueno, Hamilton, Aldo H. S. Medeiros +1
Computer Science · Mathematics · #35A01 #35B45 #35R11 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2211.16946

openalex publication_date 2022/11/30 · openalex created_date 2022/12/12 · openalex updated_date 2026/07/28

Abstract

In this paper we study a Neumann problem for the fractional Laplacian, namely \ ε^2s(- Δ)^su + u · amp;= · amp; f(u) · amp;in Ω
N_su · amp;= · amp; 0 , · amp;in ℝ^N\backslash Ω. where Ω⊂ ℝ^N is a smooth bounded domain, N>2s, s ∈ (0,1), ε > 0 is a parameter and N_s is the nonlocal normal derivative introduced by Dipierro, Ros-Oton, and Valdinoci. We establish the existence of a nonnegative, non-constant small energy solution u_ε, and we use the Moser-Nash iteration procedure to show that u_ε ∈ L^∞(Ω).

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