Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms

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 −Δmu=uq|∇u|pinRN,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 −Δmu=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.