Global L^∞ and decay estimate for fractional p-Laplacian equations in D^s,p(\R^N)

2025/07/13 by Siegfried Carl, Carl, Siegfried, Kanishka Perera +3
Engineering · Mathematics · #31C05 #35B38 #35B40 #35B45 #35B51 #35J20 #35J60 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2507.09533

openalex publication_date 2025/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we present a new global L^∞-estimate for solutions u∈ D^s,p(\R^N) of the fractional p-Laplacian equation % u∈ D^s,p(\R^N): (-Δ_p)^s u=f(x,u) \quadin \R^N, % of the form % ‖u‖_∞≤ C Φ(‖u‖_β) % for some β> p, where Φ: \R⁺→ \R⁺ is a data independent function with lim_{s→ 0⁺}Φ(s)=0. The obtained L^∞-estimate is used to prove a decay estimate based on pointwise estimates in terms of nonlinear Wolff potentials. Taking advantage of both the L^∞ and decay estimate we prove a Brezis-Nirenberg type result regarding D^s,2(\R^N) versus C_b(\R^N, 1+|x|^N-2s) local minimizers.

Citations

• Wolff potential estimates and Wiener criterion for nonlocal equations with Orlicz growth
• Sobolev versus H "older minimizers for the degenerate fractional\n p-Laplacian
• A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional p(⋅)-Laplacian
• A priori bounds and existence of solutions for some nonlocal elliptic problems
• A note on global regularity for the weak solutions of fractional\n p-Laplacian equations
• Nonlocal Harnack inequalities
• Nonlocal Harnack inequalities
• H^s versus C⁰-weighted minimizers
• Fractional Eigenvalues

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