Singular Trudinger–Moser inequality and fractional p-Laplace equations in RN. (July 2020)

• Record Type: Journal Article
• Authors: Van Thin, Nguyen
• Is Part Of: Nonlinear analysis. Volume 196(2020)
• Publication Date: 2020-07
• Subjects: primary 35A15 35A23 35J35 35J60 35R11; Integrodifferential operators -- Trudinger–Moser inequality -- Fractional p-Laplace -- Mountain Pass Theorem -- Variational principle; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.111756
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we establish a singular Trudinger–Moser inequality in R N . Applying that result, we study the existence of solution to N ∕ s -fractional Laplacian equation L N ∕ s s u ( x ) + V ( x ) | u | N s − 2 u = f ( x, u ) | x | γ, where γ ∈ [ 0, N ), 0 < s < 1, L N ∕ s s is a nonlocal fractional operator with singular kernel K : R N ∖ { 0 } → R +, f is a continuous function on R N × R which does not satisfy the Ambrosetti–Rabinowitz condition. By using Mountain Pass Theorem, we obtain the existence of solution to above equation. Furthermore, when f satisfies the Ambrosetti–Rabinowitz condition, we obtain the existence to solution of that equation with negative energy. In our best knowledge, it is the first time the existence to solution of above equation is studied.