We prove the nonexistence of solutions for a prescribed mean curvature equation { − div ( ∇ u 1 + | ∇ u | 2 ) = λ | u | p − 1 u , x ∈ B R ⊆ R n , u = 0 , x ∈ ∂ B R , when p ⩾ 1 and the positive parameter λ is small. The result extends theorems of Narukawa and Suzuki, and Finn, from the case of n = 2 , p = 1 to all n ⩾ 2 , p ⩾ 1 . Moreover, our proof is very simple and the result is not limited to positive (and negative) solutions. We also show that a similar result for positive solutions is still true if | u | p − 1 u is replaced by the exponential nonlinearity e u − 1 .