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We provide the existence of a solution for quasilinear elliptic equation −div(∞()|∇|−2∇+̃(,|∇|)∇)=()||−2+(,)+ℎ() in Ω under the Neumann boundary condition. Here, we consider the condition that ̃(,)=(−2) as →+∞ and (,)=(||−1) as ||→∞. As a special case, our result implies that the following -Laplace equation has at least one solution: −Δ=()||−2+||−2+ℎ() in Ω,/=0 on Ω for every 1<<<∞, ∈ℝ, ≠0 and ,ℎ∈∞(Ω) with ∫Ω≠0. Moreover, in the nonresonant case, that is,  is not an eigenvalue of the -Laplacian with weight , we present the existence of a solution of the above -Laplace equation for every 1<<<∞, ∈ℝ and ,ℎ∈∞(Ω)

Existence Results for Quasilinear Elliptic Equations with Indefinite Weight