We consider the Dirichlet problem for positive solutions of the equation -Δm(u)=f(u) in a bounded smooth domain Ω, with f locally Lipschitz continuous, and prove some regularity results for weak C1(Ω̄) solutions. In particular when f(s)>0 for s>0 we prove summability properties of 1/|Du|, and Sobolev's and Poincaré type inequalities in weighted Sobolev spaces with weight |Du|m-2. The point of view of considering |Du|m-2 as a weight is particularly useful when studying qualitative properties of a fixed solution. In particular, exploiting these new regularity results we can prove a weak comparison principle for the solutions and, using the well known Alexandrov-Serrin moving plane method, we then prove a general monotonicity (and symmetry) theorem for positive solutions u of the Dirichlet problem in bounded (and symmetric in one direction) domains when f(s)>0 for s>0 and m>2. Previously, results of this type in general bounded (and symmetric) domains had been proved only in the case 1<m<2. © 2004 Elsevier Inc. All rights reserved.
Damascelli, L., & Sciunzi, B. (2004). Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations. Journal of Differential Equations, 206(2), 483–515. https://doi.org/10.1016/j.jde.2004.05.012