Existence and Uniqueness for the Fractional Gelfand Equation in ℝ

2025/06/09 by Florian P. Lanz, Enno Lenzmann, Lanz, Florian P. +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.07577

openalex publication_date 2025/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove existence, symmetry and uniqueness of solutions to the fractional Gelfand equation (-Δ)^s u = e^u in ℝ with ∫_ℝ e^u dx lt; +∞ for all exponents s ∈ ((1)/(2),1). Furthermore, we show u has finite Morse index and that its linearized operator is nondegenerate. Our arguments are based on a fixed point scheme in terms of the function v= √(e^u) and we devise a nonlocal shooting method involving (locally) compact nonlinear maps. We also study existence, symmetry and uniqueness of solutions to (-Δ)^s u = K e^u in ℝ with K e^u ∈ L¹(ℝ) for a general class of positive, even and monotone-decreasing functions K > 0.

Citations

• Non-degeneracy of the bubble in a fractional and singular 1D Liouville equation
• The Calogero--Moser Derivative Nonlinear Schrödinger Equation
• Classification of finite Morse index solutions of higher-order Gelfand-Liouville equation
• Classification of stable solutions to a non-local Gelfand-Liouville equation
• A Liouville type theorem for fractional elliptic equation with exponential nonlinearity
• Reverse Hardy-Littlewood-Sobolev inequalities
• Entire s-harmonic functions are affine
• Functionals for Multilinear Fractional Embedding
• Reversed Hardy-Littewood-Sobolev inequality
• The extremal solution for the fractional Laplacian
• Uniqueness of Radial Solutions for the Fractional Laplacian
• Uniqueness theorems through the method of moving spheres

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• Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions