Existence of Positive Solutions for Generalized Fractional Brézis-Nirenberg Problem

2024/06/09 by Rohit Kumar, Abhishek Sarkar, Kumar, Rohit +1 · 1 citation
Mathematics · #35B09 #35R11 #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2406.05793

openalex publication_date 2024/06/09 · openalex created_date 2024/06/12 · openalex updated_date 2026/07/28

Abstract

In this article, we study the fractional Brézis-Nirenberg type problem on whole domain ℝ^N associated with the fractional p-Laplace operator. To be precise, we want to study the following problem: (-Δ)_p^su - λw |u|^p-2u= |u|^p_s^*-2u in ~D^s,p(ℝ^N), where s∈ (0,1),~p ∈ (1,(N)/(s)), ~p_s^*= (Np)/(N-sp) and the operator (-Δ)_p^s is the fractional p-Laplace operator. The space D^s,p(ℝ^N) is the completion of C_c^∞(ℝ^N) with respect to the Gaglairdo semi-norm. In this article, we prove the existence of a positive solution to this problem by allowing the Hardy weight w to change its sign.

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