Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms. (July 2022)

• Record Type: Journal Article
• Authors: Chang, Caihong; Hu, Bei; Zhang, Zhengce
• Is Part Of: Nonlinear analysis. Volume 220(2022)
• Publication Date: 2022-07
• Subjects: 35J60 -- 35J92 -- 35J70; m-Laplacian -- Gradient terms -- Bernstein-type estimates -- Liouville-type theorems -- Harnack inequalities -- A priori estimates -- Blow-up argument; 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.2022.112873
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This paper is concerned with two properties of positive weak solutions of quasilinear elliptic equations with nonlinear gradient terms. First, we show a Liouville-type theorem for positive weak solutions of the equation involving the m -Laplacian operator − Δ m u = u q | ∇ u | p in R N, where N ⩾ 1, m > 1 and p, q ⩾ 0 . The technique of Bernstein gradient estimates is utilized to study the case p < m . Moreover, a Liouville-type theorem for supersolutions under subcritical range of exponents q ( N − m ) + p ( N − 1 ) < N ( m − 1 ) is also established. Then, we use a degree argument to obtain the existence of positive weak solutions for a nonlinear Dirichlet problem of the type − Δ m u = f ( x, u, ∇ u ), with f satisfying certain structure conditions. Our proof is based on a priori estimates, which will be accomplished by using a blow-up argument together with the Liouville-type theorem in the half-space. As another application, some new Harnack inequalities are proved.