Existence Results for Quasilinear Elliptic Equations with Indefinite Weight

We provide the existence of a solution for quasilinear elliptic equation − d i v ( ð ‘Ž ∞ ( ð ‘¥ ) | ∇ ð ‘¢ | ð ‘ âˆ’ 2 ∇ ð ‘¢ + ̃ ð ‘Ž ( ð ‘¥ , | ∇ ð ‘¢ | ) ∇ ð ‘¢ ) = 𠜆 ð ‘š ( ð ‘¥ ) | ð ‘¢ | ð ‘ âˆ’ 2 ð ‘¢ + ð ‘“ ( ð ‘¥ , ð ‘¢ ) + â„Ž ( ð ‘¥ ) in Ω under the Neumann boundary condition. Here, we consider the condition that ̃ ð ‘Ž ( ð ‘¥ , ð ‘¡ ) = ð ‘œ ( ð ‘¡ ð ‘ âˆ’ 2 ) as ð ‘¡ → + ∞ and ð ‘“ ( ð ‘¥ , ð ‘¢ ) = ð ‘œ ( | ð ‘¢ | ð ‘ âˆ’ 1 ) as | ð ‘¢ | → ∞ . As a special case, our result implies that the following ð ‘ -Laplace equation has at least one solution: − Δ ð ‘ ð ‘¢ = 𠜆 ð ‘š ( ð ‘¥ ) | ð ‘¢ | ð ‘ âˆ’ 2 ð ‘¢ + 𠜇 | ð ‘¢ | ð ‘Ÿ − 2 ð ‘¢ + â„Ž ( ð ‘¥ ) in Ω , 𠜕 ð ‘¢ / 𠜕 𠜈 = 0 on 𠜕 Ω for every 1 < ð ‘Ÿ < ð ‘ < ∞ , 𠜆 ∈ â„ , 𠜇 ≠0 and ð ‘š , â„Ž ∈ ð ¿ âˆž ( Ω ) with ∫ Ω ð ‘š ð ‘‘ ð ‘¥ ≠0 . Moreover, in the nonresonant case, that is, 𠜆 is not an eigenvalue of the ð ‘ -Laplacian with weight ð ‘š , we present the existence of a solution of the above ð ‘ -Laplace equation for every 1 < ð ‘Ÿ < ð ‘ < ∞ , 𠜇 ∈ â„ and ð ‘š , â„Ž ∈ ð ¿ âˆž ( Ω ) .