We provide the existence of a solution for quasilinear elliptic equation in Ω under the Neumann boundary condition. Here, we consider the condition that as t → +∞ and f(x, u) = o(|u|p−1) as |u | → ∞. As a special case, our result implies that the following p‐Laplace equation has at least one solution: −Δpu = λm(x) | u|p−2u + μ | u|r−2u + h(x) in Ω, ∂u/∂ν = 0 on ∂Ω for every 1 < r < p < ∞, λ ∈ ℝ, μ ≠ 0 and m, h ∈ L∞(Ω) with ∫Ωm dx ≠ 0. Moreover, in the nonresonant case, that is, λ is not an eigenvalue of the p‐Laplacian with weight m, we present the existence of a solution of the above p‐Laplace equation for every 1 < r < p < ∞, μ ∈ ℝ and m, h ∈ L∞(Ω).