We consider the Dirichlet problem for positive solutions of the equation $ -\Delta_m (u) = f(u) \; $ in a bounded, smooth domain $\, \Omega $, with $f$ positive and locally Lipschitz continuous. We prove a weak maximum principle in small domains for the linearized operator that we exploit to prove a weak maximum principle for the linearized operator. We then consider the case $f(s)=s^q$ and prove a nondegeneracy result in weighted Sobolev spaces.
"A weak maximum principle for the linearized operator of $m$-Laplace equations with applications to a nondegeneracy result." Adv. Differential Equations 10 (2) 223 - 240, 2005.