We consider the problem − div ( | ∇ u | p − 2 ∇ u ) = a ( x ) ( u m + λ u n ) , x ∈ ℝ N , N ≥ 3, where 0 < m < p − 1 < n , a ( x ) ≥ 0 , a ( x ) is not identically zero. Under the condition that a ( x ) satisfies (H), we show that there exists λ 0 > 0 such that the above-mentioned equation admits at least one solution for all λ ∈ ( 0 , λ 0 ) . This extends the results of Laplace equation to the case of p -Laplace equation.