Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms. (July 2022) - Chang, Caihong; Hu, Bei; Zhang, Zhengce
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.