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We consider a quasilinear elliptic equation involving a first order term,under zero Dirichlet boundary condition in half spaces. We prove that anypositive solution is monotone increasing w.r.t. the direction orthogonal to theboundary. The main ingredient in the proof is a new comparison principle inunbounded domains. As a consequence of our analysis, we also obtain some newLiouville type theorems.

Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces