Existence of Solutions for Some Quasilinear Degenerate Elliptic Inclusions in Weighted Sobolev Spaces. (2nd January 2016)

• Record Type: Journal Article
• Authors: Cheng, Yi; Li, Cuiying
• Is Part Of: Numerical functional analysis and optimization. Volume 37:Number 1(2016)
• Page Start: 40
• Page End: 50
• Publication Date: 2016-01-02
• Subjects: Degenerate elliptic inclusion -- extremal solution -- Muckenhoupt weights -- weighted Sobolev space; 34A60 -- 35J60; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2015.1078811
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: We consider the Dirichlet problem for a class of quasilinear degenerate elliptic inclusions of the form −div(𝒜( x, u, ∇ u )) + f ( x ) g ( u ) ∈ H ( x, u, ∇ u ), where 𝒜( x, u, ∇ u ) is allowed to be degenerate. Without the general assumption that the multivalued nonlinearity is characterized by Clarke's generalized gradient of some locally Lipschitz functions, we prove the existence of bounded solutions in weighed Sobolev space with the superlinear growth imposed on the nonlinearity g and the multifunction H ( x, u, ∇ u ) by using the Leray-Schauder fixed point theorem. Furthermore, we investigate the existence of extremal solutions and prove that they are dense in the solutions of the original system. Subsequently, a quasilinear degenerate elliptic control problem is considered and the existence theorem based on the proven results is obtained.