Extensions of a theorem of Cauchy–Liouville

We deal with the equations Δ p u + f ( u ) = 0 and Δ p u + ( p − 1 ) g ( u ) | ∇ u | p + f ( u ) = 0 in R N , where g ( t ) is a continuous function in ( 0 , ∞ ) , p > 1 and f ( t ) is a smooth function for t > 0 . Under appropriate conditions on g and f we show that the corresponding equation cannot have nontrivial non-negative entire solutions.