Existence and concentration of positive ground states for a 1-Laplacian problem in RN. (February 2020)

• Record Type: Journal Article
• Authors: Che, Guofeng; Shi, Hongxia; Wang, Zewei
• Is Part Of: Applied mathematics letters. Volume 100(2020)
• Publication Date: 2020-02
• Subjects: 1-Laplacian operator -- Bounded variation functions -- Concentration-compactness principle; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2019.106045
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: This paper is concerned with the following quasilinear elliptic problem: − ε Δ 1 u + V ( x ) u | u | = K ( x ) | u | p − 1 u, in R N, u ∈ B V ( R N ), where 0 < p < 1 N − 1, N ≥ 2 and ε > 0 is a small parameter. Under some mild conditions on the nonnegative functions V ( x ) and K ( x ), we establish the existence of a ground state solution u ε for the above problem. Moreover, u ε concentrates on the intersection set of global minimum points of V ( x ) and maximum points of K ( x ) . The methods are based on the Nehari manifold technique and the Concentration-Compactness Principle of Lions.